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Tocino</a><sup class="u-js-hide"><a href="#Aff3">3</a></sup> & </li><li class="c-article-author-list__show-more" aria-label="Show all 4 authors for this article" title="Show all 4 authors for this article">…</li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-A_-Viruel-Aff2" data-author-popup="auth-A_-Viruel-Aff2" data-author-search="Viruel, A.">A. 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We show that if a field <span class="mathjax-tex">$\Bbbk $</span> contains sufficiently many elements (for example if <span class="mathjax-tex">$\Bbbk $</span> is infinite) then every finite group <i>G</i> is isomorphic to <span class="mathjax-tex">${\text {Aut}}(X)$</span> where <i>X</i> is a finite-dimensional absolutely simple Cayley evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra.</p></div></div></section> <div data-test="cobranding-download"> </div> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w92h120/springer-static/cover-hires/book/978-3-030-35256-1?as=webp" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/978-3-030-35256-1_2?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1007/978-3-030-35256-1_2">Power-Associative Evolution Algebras </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Chapter</span> <span class="c-article-meta-recommendations__date">© 2020</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s00013-018-1178-9?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1007/s00013-018-1178-9">Evolving groups </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">13 April 2018</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1134%2FS0037446622050184/MediaObjects/11202_2022_1238_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1134/S0037446622050184?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1134/S0037446622050184">On the Hilbert Evolution Algebras of a Graph </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">01 September 2022</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1732635770, embedded_user: 'null' } }); </script> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=13398" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>The question of whether any finite group may be realised as the full automorphism group of a finite-dimensional simple algebra (over an algebraically closed field <span class="mathjax-tex">$\Bbbk $</span> of characteristic zero) was raised by Popov in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Popov, V.L.: An analogue of M. Artin’s conjecture on invariants for non-associative algebras. Am. Math. Soc. Transl. 169, 121–143 (1995)" href="/article/10.1007/s13398-023-01414-w#ref-CR13" id="ref-link-section-d2809547e614">13</a>]. Years later, in the celebrated paper by Gordeev and Popov [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Gordeev, N.L., Popov, V.L.: Automorphism groups of finite dimensional simple algebras. Ann. Math. 158, 1041–1065 (2003)" href="/article/10.1007/s13398-023-01414-w#ref-CR9" id="ref-link-section-d2809547e617">9</a>], a positive answer was given in a more general setting: if <span class="mathjax-tex">$\Bbbk $</span> is a field containing sufficiently many elements, every linear algebraic <span class="mathjax-tex">$\Bbbk $</span>-group is isomorphic to the full automorphism group of a finite-dimensional simple <span class="mathjax-tex">$\Bbbk $</span>-algebra (which is neither associative nor commutative).</p><p>Using techniques from Rational Homotopy Theory, in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Costoya, C., Méndez, D., Viruel, A.: Realisability problem in arrow categories. Collect. Math. 71, 383–405 (2020)" href="/article/10.1007/s13398-023-01414-w#ref-CR6" id="ref-link-section-d2809547e678">6</a>, Proposition 4.2, Corollary 4.11(2)] it was shown that every group <i>G</i> is the full automorphism group of an infinite number of non-isomorphic differential graded <span class="mathjax-tex">${\mathbb {Q}}$</span>-algebras <span class="mathjax-tex">$M_n,\; n \ge 1$</span> (that are associative and commutative in the graded sense). Moreover, if the group <i>G</i> is finite, then for every integer <span class="mathjax-tex">$n \ge 1$</span>, <span class="mathjax-tex">$M_n$</span> is an elliptic algebra, which means that it is finitely generated with finite-dimensional cohomology over <span class="mathjax-tex">${\mathbb {Q}}.$</span></p><p>In this paper, we place ourselves in the framework of evolution algebras, which are commutative and non-associative algebras. Recall that a <span class="mathjax-tex">$\Bbbk $</span>-algebra <i>X</i> is called absolutely simple, if for every field extension <span class="mathjax-tex">$\mathbb {F}/\Bbbk $</span>, the <span class="mathjax-tex">$\mathbb {F}$</span>-algebra <span class="mathjax-tex">$X_{{\mathbb {F}}}=X\otimes _\Bbbk {{\mathbb {F}}}$</span> is simple. Then our main result is:</p> <h3 class="c-article__sub-heading" id="FPar1">Theorem A</h3> <p>Let <i>G</i> be a finite group and let <span class="mathjax-tex">$\Bbbk $</span> be a not necessarily finite field with multiplicative group of order <span class="mathjax-tex">$|\Bbbk ^*| \ge 2|G|$</span>. Then, there exists an absolutely simple evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra <i>X</i> such that <span class="mathjax-tex">${\text {Aut}}(X_{{\mathbb {F}}})\cong G$</span> for every field extension <span class="mathjax-tex">$\mathbb {F}/\Bbbk $</span>.</p> <p>The proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar1">A</a> is postponed to Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s13398-023-01414-w#Sec4">4</a> and relies on the <i>good</i> properties of Cayley evolution algebras, which are new objects introduced in this work (see Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar6">3.1</a>). They are constructed out of any finite-dimensional associative <span class="mathjax-tex">$\Bbbk $</span>-algebra <span class="mathjax-tex">${\mathcal {A}}$</span>, <span class="mathjax-tex">$\Bbbk $</span> an arbitrary field. In the special case that <span class="mathjax-tex">${\mathcal {A}}= \Bbbk [G]$</span>, a group algebra with <i>G</i> finite group, the associated Cayley evolution algebras will be closely related to classical Cayley graphs.</p><p>We end this section by pointing out that our main result is an improvement of our previous result [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Costoya, C., Ligouras, P., Tocino, A., Viruel, A.: Regular evolution algebras are universally finite. Proc. Am. Math. Soc. 150(3), 919–925 (2022)" href="/article/10.1007/s13398-023-01414-w#ref-CR5" id="ref-link-section-d2809547e1200">5</a>, Theorem 1.1] where finite groups were realised by regular, but not simple (see Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar13">3.6</a>), evolution algebras (see also [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Sriwongsa, S., Zou, Y.M.: On automorphism groups of idempotent evolution algebras. Linear Algebra Appl. 641, 143–155 (2022)" href="/article/10.1007/s13398-023-01414-w#ref-CR15" id="ref-link-section-d2809547e1206">15</a>] for an independent proof in char <span class="mathjax-tex">$\Bbbk =0$</span>).</p></div></div></section><section data-title="Background on evolution algebras"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Background on evolution algebras</h2><div class="c-article-section__content" id="Sec2-content"><p>We present here the definitions on evolution algebras that are needed in the following sections. We mainly follow the notation in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Cabrera C.Y.: Evolution algebras. Ph.D. thesis, University of Málaga, Spain (2016)" href="/article/10.1007/s13398-023-01414-w#ref-CR3" id="ref-link-section-d2809547e1242">3</a>], although the reader is encouraged to also check [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Tian, J.P.: Evolution Algebras and Their Applications. Lecture Notes in Mathematics, vol. 1921. Springer, Berlin (2008)" href="/article/10.1007/s13398-023-01414-w#ref-CR16" id="ref-link-section-d2809547e1245">16</a>], a classical reference on the subject. Other properties of evolution algebras appear in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Ahmed, H., Bekbaev, U., Rakhimov, I.: On classification of 
 
 
 
 $$2$$
 
 
 2
 
 
 -dimensional evolution algebras and its applications. In: 5th International Conference on Mathematical Applications in Engineering 30–31 October 2019, Putrajaya, Malaysia, vol. 1489, pp. 012001. IOP Publishing Ltd (2020)" href="/article/10.1007/s13398-023-01414-w#ref-CR1" id="ref-link-section-d2809547e1248">1</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Alarafeen, A., Qaralleh, I., Ahmad, A.: Properties of nilpotent evolution algebras with no maximal nilindex. Eur. J. Pure Appl. Math. 14(1), 278–300 (2021)" href="/article/10.1007/s13398-023-01414-w#ref-CR2" id="ref-link-section-d2809547e1251">2</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Mukhamedov, F., Khakimov, O., Omirov, B., Qaralleh, I.: Derivations and automorphisms of nilpotent evolution algebras with maximal nilindex. J. Algebra Appl. 18(12), 1950233 (2019)" href="/article/10.1007/s13398-023-01414-w#ref-CR12" id="ref-link-section-d2809547e1254">12</a>].</p><p>Let <span class="mathjax-tex">$\Bbbk $</span> be a field. An automorphism of <span class="mathjax-tex">$\Bbbk $</span>-algebra is a linear isomorphism which commutes with the multiplication of the algebra. In this work we consider only finite-dimensional algebras. We now recall the definition of an evolution algebra [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Tian, J.P.: Evolution Algebras and Their Applications. Lecture Notes in Mathematics, vol. 1921. Springer, Berlin (2008)" href="/article/10.1007/s13398-023-01414-w#ref-CR16" id="ref-link-section-d2809547e1296">16</a>, Definition 1], [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Cabrera C.Y.: Evolution algebras. Ph.D. thesis, University of Málaga, Spain (2016)" href="/article/10.1007/s13398-023-01414-w#ref-CR3" id="ref-link-section-d2809547e1299">3</a>, Definitions 1.2.1]:</p> <h3 class="c-article__sub-heading" id="FPar2">Definition 2.1</h3> <p>Let <i>X</i> be a <span class="mathjax-tex">$\Bbbk $</span>-algebra. We say that <i>X</i> is an <i>evolution </i><span class="mathjax-tex">$\Bbbk $</span>-<i>algebra</i> if <i>X</i> admits a basis <span class="mathjax-tex">$B = \{b_i\, |\, i \in \Lambda \}$</span> such that <span class="mathjax-tex">$b_i b_j = 0$</span> for <span class="mathjax-tex">$i\ne j$</span>. Such a basis <i>B</i> is called a <i>natural basis</i>.</p> <p>Observe that given an evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra <i>X</i>, it may admit more than one natural basis (see for example [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Cabrera C.Y.: Evolution algebras. Ph.D. thesis, University of Málaga, Spain (2016)" href="/article/10.1007/s13398-023-01414-w#ref-CR3" id="ref-link-section-d2809547e1501">3</a>, Example 1.6.3]). On the other hand, for a given <span class="mathjax-tex">$\Bbbk $</span>-vector space <i>X</i> spanned by a basis <span class="mathjax-tex">$B = \{b_i\, |\, i \in \Lambda \}$</span>, an evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra with natural basis <i>B</i> is completely determined by just giving <span class="mathjax-tex">$b_i^2:=b_i b_i\in X$</span> as a linear combination of elements in <i>B</i>, for every <span class="mathjax-tex">$i\in \Lambda $</span>, and declaring <span class="mathjax-tex">$b_ib_j=0$</span> for <span class="mathjax-tex">$i \ne j$</span>. This motivates the following definition [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Cabrera C.Y.: Evolution algebras. Ph.D. thesis, University of Málaga, Spain (2016)" href="/article/10.1007/s13398-023-01414-w#ref-CR3" id="ref-link-section-d2809547e1736">3</a>, Definition 1.2.1]:</p> <h3 class="c-article__sub-heading" id="FPar3">Definition 2.2</h3> <p>Let <i>X</i> be an evolution algebra with a fixed natural basis <i>B</i>. The scalars <span class="mathjax-tex">$\omega _{ki} \in \Bbbk $</span> such that <span class="mathjax-tex">$b_i^2:=b_i b_i=\sum _{k\in \Lambda }\omega _{ki}b_k$</span> are called the <i>structure constants</i> of <i>X</i> relative to <i>B</i>, and the matrix <span class="mathjax-tex">$M_B(X):= (\omega _{ki})$</span> is said to be the <i>structure matrix</i> of <i>X</i> relative to <i>B</i>.</p> <h3 class="c-article__sub-heading" id="FPar4">Definition 2.3</h3> <p>An evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra <i>X</i> is called <i>regular</i> (or perfect, or idempotent) if <span class="mathjax-tex">$X=X^2$</span>. An evolution algebra <i>X</i> is called <i>simple</i> if <span class="mathjax-tex">$X^2\ne 0$</span> and 0 is the only proper ideal.</p> <p>Notice that an evolution algebra <i>X</i> is regular if and only if for any natural basis <i>B</i>, the structure matrix <span class="mathjax-tex">$M_B(X)$</span> is a regular (or invertible) matrix. It is also clear that if <i>X</i> simple then <i>X</i> is regular, but the converse is not always true (see Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar11">3.5</a>).</p><p>Every evolution algebra has a directed graph and a weighted (or coloured) graph attached depending on the chosen natural basis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Tian, J.P.: Evolution Algebras and Their Applications. Lecture Notes in Mathematics, vol. 1921. Springer, Berlin (2008)" href="/article/10.1007/s13398-023-01414-w#ref-CR16" id="ref-link-section-d2809547e2103">16</a>, Definition 15], [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Elduque, A., Labra, A.: Evolution algebras and graphs. J. Algebra Appl. 14(7), 1550103 (2015)" href="/article/10.1007/s13398-023-01414-w#ref-CR7" id="ref-link-section-d2809547e2106">7</a>, Definition 2.2]:</p> <h3 class="c-article__sub-heading" id="FPar5">Definition 2.4</h3> <p>Let <i>X</i> be an evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra, and <span class="mathjax-tex">$B=\{b_1,\ldots ,b_n\}$</span> its natural basis. The directed graph <span class="mathjax-tex">$\Gamma (X,B)=(V,E)$</span> with set of vertices <span class="mathjax-tex">$V=\{1,\ldots ,n\}$</span> and set of edges <span class="mathjax-tex">$E=\{(i,j)\in V\times V: \omega _{ij}\ne 0\}$</span> is called the directed graph attached to the evolution algebra <i>X</i> relative to <i>B</i>. The directed graph <span class="mathjax-tex">$\Gamma ^w(X,B)=(V,E,w)$</span> with <span class="mathjax-tex">$\Gamma (X,B)=(V,E)$</span> and weight function <span class="mathjax-tex">$w:E\rightarrow \Bbbk $</span>, given by <span class="mathjax-tex">$w(i,j)=\omega _{ij}$</span>, is called the weighted graph attached to the evolution algebra <i>X</i> relative to <i>B</i>.</p> </div></div></section><section data-title="The Cayley evolution algebra of a $\Bbbk $-algebra"><div class="c-article-section" id="Sec3-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec3"><span class="c-article-section__title-number">3 </span>The Cayley evolution algebra of a <span class="mathjax-tex">$\Bbbk $</span>-algebra</h2><div class="c-article-section__content" id="Sec3-content"><p>We now explain how, to a given associative <span class="mathjax-tex">$\Bbbk $</span>-algebra <span class="mathjax-tex">$\mathcal {A}$</span>, we can associate an evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra that takes into account the inner product of <span class="mathjax-tex">$\mathcal {A}$</span>:</p> <h3 class="c-article__sub-heading" id="FPar6">Definition 3.1</h3> <p>Let <span class="mathjax-tex">$\mathcal {A}$</span> be a finite-dimensional associative <span class="mathjax-tex">$\Bbbk $</span>-algebra with basis <span class="mathjax-tex">$B=\{b_i\,:\,i\in \Lambda \}$</span> together with a function <span class="mathjax-tex">$f:B\rightarrow \Bbbk $</span> (a set-theoretical map). We define the <i>Cayley evolution </i> <span class="mathjax-tex">$\Bbbk $</span><i>-algebra associated to </i><i>f</i>, that we denote by <span class="mathjax-tex">$\textrm{Cay}(f)$</span>, as the evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra with <span class="mathjax-tex">$\textrm{Cay}(f)\cong \mathcal {A}$</span>, as a <span class="mathjax-tex">$\Bbbk $</span>-vector space, furnished with natural basis <i>B</i> and structure constants given by <span class="mathjax-tex">$b_i\cdot b_i=\sum _{b_j\in B}f(b_j)b_ib_j$</span>, where <span class="mathjax-tex">$x\cdot y$</span> denotes the product in <span class="mathjax-tex">$\textrm{Cay}(f)$</span> and <i>xy</i> the inner product in <span class="mathjax-tex">$\mathcal {A}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar7">Remark 3.2</h3> <p>Observe that fixing a function <span class="mathjax-tex">$f:B\rightarrow \Bbbk $</span> is equivalent to choosing a vector <span class="mathjax-tex">$w=\sum f(b_i)b_i$</span>. Therefore, the Cayley evolution algebra associated to <i>f</i> could also be defined as the Cayley evolution algebra associated to a vector <span class="mathjax-tex">$w\in \mathcal {A}$</span> with <span class="mathjax-tex">$\textrm{Cay}(w)\cong \mathcal {A}$</span>, as <span class="mathjax-tex">$\Bbbk $</span>-vector space, furnished with natural basis <i>B</i> and structure constants given by <span class="mathjax-tex">$b_i\cdot b_i=\sum _{b_j\in B}w^*(b_j)b_ib_j$</span>, where <span class="mathjax-tex">$w^*$</span> is the dual of <i>w</i>.</p> <h3 class="c-article__sub-heading" id="FPar8">Remark 3.3</h3> <p>Given a finite-dimensional associative <span class="mathjax-tex">$\Bbbk $</span>-algebra <span class="mathjax-tex">$\mathcal {A}$</span> with basis <i>B</i>, the isomorphism type of the evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra <span class="mathjax-tex">$\textrm{Cay}(f)$</span> strongly depends on the choice of <span class="mathjax-tex">$f:B\rightarrow \Bbbk $</span>. For example, if <span class="mathjax-tex">$\mathcal {A}$</span> is a non-degenerate evolution algebra with natural basis <i>B</i>, then the constant map given by <span class="mathjax-tex">$f_0(b_i)=0$</span> converts <span class="mathjax-tex">$\textrm{Cay}(f_0)$</span> into a degenerated evolution algebra, whereas the constant map given by <span class="mathjax-tex">$f_1(b_i)=1$</span> transforms <span class="mathjax-tex">$\textrm{Cay}(f_1)\cong \mathcal {A}$</span>. A more elaborated example is constructed in Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar27">4.9</a> to illustrate how the isomorphism type of the Cayley evolution algebra depends on the choice of the basis.</p> <p>From now on, let us fix a unital finite-dimensional associative <span class="mathjax-tex">$\Bbbk $</span>-algebra <span class="mathjax-tex">$\mathcal {A}$</span>, with basis <i>B</i>, and a function <span class="mathjax-tex">$f:B\rightarrow \Bbbk $</span>. We now prove several properties of the Cayley evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra associated to <i>f</i>, <span class="mathjax-tex">$\textrm{Cay}(f)$</span>. First we show how units in <span class="mathjax-tex">$\mathcal {A}$</span> give rise to automorphisms of <span class="mathjax-tex">$\textrm{Cay}(f)$</span>.</p> <h3 class="c-article__sub-heading" id="FPar9">Proposition 3.4</h3> <p>If <span class="mathjax-tex">$x\in \mathcal {A}$</span> is a unit such that <span class="mathjax-tex">$xB=B$</span> then the left-multiplication by <i>x</i> induces <span class="mathjax-tex">$\psi _x\in {\text {Aut}}\big (\textrm{Cay}(f)\big )$</span>. If moreover <i>B</i> consists of units in <span class="mathjax-tex">$\mathcal {A}$</span>, then <span class="mathjax-tex">$\psi _x$</span> acts freely on <i>B</i> for <span class="mathjax-tex">$x \ne 1$</span>.</p> <h3 class="c-article__sub-heading" id="FPar10">Proof</h3> <p>Given a unit <span class="mathjax-tex">$x\in \mathcal {A}$</span>, we construct the map <span class="mathjax-tex">$\psi _x:\textrm{Cay}(f)\rightarrow \textrm{Cay}(f)$</span> given by <span class="mathjax-tex">$\psi _x(a)=xa$</span>. Observe that <span class="mathjax-tex">$\psi _x$</span> maps the natural basis <i>B</i> to itself since <span class="mathjax-tex">$\psi _x(B)=xB=B$</span>. Hence it is an automorphism of <span class="mathjax-tex">$\textrm{Cay}(f)$</span> as <span class="mathjax-tex">$\Bbbk $</span>–vector space. Now, to check that <span class="mathjax-tex">$\psi _x$</span> is an automorphism of evolution algebras, on the one hand we have that <span class="mathjax-tex">$\psi _x(b_i\cdot b_i)=\psi _x(\sum _{b_j\in B}f(b_j)b_ib_j)=\sum _{b_j\in B}f(b_j)x(b_ib_j)=\sum _{b_j\in B}f(b_j)(x b_i)b_j= \sum _{b_j\in B}f(b_j)\psi _x(b_i)b_j=\psi _x(b_i)\cdot \psi _x(b_i)$</span>. On the other hand, for <span class="mathjax-tex">$b_i \ne b_j,$</span> we have that <span class="mathjax-tex">$\psi _x(b_i\cdot b_j)=\psi _x(0)=x0=0 $</span> and we are going to prove that <span class="mathjax-tex">$\psi _x(b_i)\cdot \psi _x(b_j)$</span> is also zero. Suppose that it is not, hence <span class="mathjax-tex">$ \psi _x(b_i)\cdot \psi _x(b_j)=xb_i\cdot xb_j \ne 0$</span>, and since <span class="mathjax-tex">$xb_i$</span> and <span class="mathjax-tex">$xb_j $</span> are elements in the natural basis <i>B</i>, they must be equal. But <i>x</i> is a unit in <span class="mathjax-tex">$\mathcal {A}$</span>, which leads to the contradiction <span class="mathjax-tex">$b_i = b_j$</span>.</p> <p>Now, suppose that <i>B</i> consists of units in <span class="mathjax-tex">$\mathcal {A}$</span>. If <span class="mathjax-tex">$\psi _x$</span> fixes an element <span class="mathjax-tex">$b \in B$</span>, then <span class="mathjax-tex">$\psi _x(b)=xb= b$</span>, so we obtain that <span class="mathjax-tex">$x=1$</span> since <i>b</i> is a unit. <span class="mathjax-tex">$\square $</span></p> <p>Let us define <span class="mathjax-tex">${\text {supp}}(f: B \rightarrow \Bbbk ):=\{b\in B:f(b)\ne 0\}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar11">Proposition 3.5</h3> <p>Suppose that <i>B</i> consists of units in <span class="mathjax-tex">$\mathcal {A}$</span>, and that <span class="mathjax-tex">${\text {supp}}(f)$</span> generates <span class="mathjax-tex">$\mathcal {A}$</span> as an algebra. If the Cayley evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra associated to <i>f</i>, <span class="mathjax-tex">$\textrm{Cay}(f)$</span>, is regular, then <span class="mathjax-tex">$\textrm{Cay}(f)$</span> is also simple.</p> <h3 class="c-article__sub-heading" id="FPar12">Proof</h3> <p>Suppose that <span class="mathjax-tex">$\textrm{Cay}(f)$</span> is regular but not simple. Then, according to [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Cabrera, C.Y., Siles, M.M., Velasco, M.V.: Evolution algebras of arbitrary dimension and their decompositions. Linear Algebra Appl. 495, 122–162 (2016)" href="/article/10.1007/s13398-023-01414-w#ref-CR4" id="ref-link-section-d2809547e5480">4</a>, Corollary 4.6], the elements in <i>B</i> can be reordered, namely <span class="mathjax-tex">$B=\{b_{1},b_{2},\ldots ,b_{r},\ldots ,b_n\}$</span>, <span class="mathjax-tex">$1\le r<n$</span>, such that the structure matrix of <span class="mathjax-tex">$\textrm{Cay}(f)$</span> becomes</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} M_B(\textrm{Cay}(f))=\left( \begin{array}{ccc|ccc} * &{} \ldots &{} * &{} * &{} \ldots &{} *\\ \vdots &{} r \times r &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ * &{} \ldots &{} * &{} * &{} \ldots &{} *\\ \hline &{} &{} &{} * &{} \ldots &{} *\\ &{} \text {0}&{} &{} \vdots &{} \ddots &{} \vdots \\ &{} &{} &{} * &{} \ldots &{} *\\ \end{array} \right) . \end{aligned}$$</span></div></div><p>Let <span class="mathjax-tex">$S={\text {supp}}(f)$</span> and consider an element <span class="mathjax-tex">$b\in B\smallsetminus \{b_1,\ldots ,b_r\}$</span>. Since <i>S</i> generates <span class="mathjax-tex">$\mathcal {A}$</span> as algebra, and <i>B</i> consists of units in <span class="mathjax-tex">$\mathcal {A}$</span>, the product <span class="mathjax-tex">$b_1^{-1}b=\sum _k \lambda _k \prod _{i=1}^{l_k}s_{i,k}$</span>, where <span class="mathjax-tex">$\lambda _k\in \Bbbk $</span> and <span class="mathjax-tex">$s_{i,k}\in S$</span>. Then, <span class="mathjax-tex">$b = b_1b_1^{-1}b = b_1\big (\sum _k \lambda _k \prod _{i=1}^{l_k}s_{i,k}\big ) = \sum _k \lambda _k b_1\big (\prod _{i=1}^{l_k}s_{i,k}\big ).$</span></p> <p>We claim that <span class="mathjax-tex">$b_1\big (\prod _{i=1}^{l_k}s_{i,k}\big ) \in Span(\{b_1,\ldots ,b_r\})$</span>. Indeed, for <span class="mathjax-tex">$j\in \{1,\ldots ,r\}$</span>,</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} b_j\cdot b_j=\sum _{b_k\in B}f(b_k)b_jb_k=\sum _{s\in S}f(s)b_js, \end{aligned}$$</span></div></div><p>thus, by the shape of <span class="mathjax-tex">$M_B(\textrm{Cay}(f))$</span>, we conclude that <span class="mathjax-tex">$b_js\in Span ( \{b_1,\ldots ,b_r\} )$</span> for all <span class="mathjax-tex">$j\in \{1,\ldots ,r\}$</span> and for all <span class="mathjax-tex">$s\in S$</span>. Even more, given <span class="mathjax-tex">$\prod _{i=1}^{l_k}s_{i,k}$</span>, for <span class="mathjax-tex">$s_{i,k}\in S$</span>, an inductive argument shows that <span class="mathjax-tex">$b_j\big (\prod _{i=1}^{l_k}s_{i,k}\big )\in Span(\{b_1,\ldots ,b_r\})$</span> for all <span class="mathjax-tex">$j\in \{1,\ldots ,r\}$</span>. In particular, we obtain that <span class="mathjax-tex">$b \in Span(\{b_1,\ldots ,b_r\}).$</span> This leads to a contradiction as <i>B</i> is a linearly independent set, hence <span class="mathjax-tex">$\textrm{Cay}(f)$</span> is simple.</p> <p> <span class="mathjax-tex">$\square $</span> </p> <h3 class="c-article__sub-heading" id="FPar13">Remark 3.6</h3> <p>As we mentioned in Introduction, the evolution algebras obtained in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Costoya, C., Méndez, D., Viruel, A.: Realisability problem in arrow categories. Collect. Math. 71, 383–405 (2020)" href="/article/10.1007/s13398-023-01414-w#ref-CR6" id="ref-link-section-d2809547e7405">6</a>] are regular but not simple. Recall that associated to a simple graph <span class="mathjax-tex">${\mathcal {G}} = (V,E)$</span>, we construct an evolution algebra <span class="mathjax-tex">${\mathcal {X}} (G)$</span> over any field <span class="mathjax-tex">$\Bbbk $</span> with natural basis <span class="mathjax-tex">$B = \{b_v,\, v \in V\} \cup \{b_e,\,e \in E\}$</span> (see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Costoya, C., Méndez, D., Viruel, A.: Realisability problem in arrow categories. Collect. Math. 71, 383–405 (2020)" href="/article/10.1007/s13398-023-01414-w#ref-CR6" id="ref-link-section-d2809547e7570">6</a>, Definition 3.2]). In particular, the multiplication for the elements associated to the vertices of the graph is given by <span class="mathjax-tex">$b_v^2 = b_v,$</span> for all <span class="mathjax-tex">$v \in V$</span>. Hence, for <span class="mathjax-tex">$I_r$</span> the identity matrix of order <span class="mathjax-tex">$r= |V|$</span>, and <span class="mathjax-tex">$I_s$</span> the identity matrix of order <span class="mathjax-tex">$s=|E|$</span>, the structure matrix of <span class="mathjax-tex">${\mathcal {X}} (G)$</span> becomes</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} M_B({\mathcal {X}} (G))=\left( \begin{array}{ccc|ccc} &{} &{} &{} * &{} \ldots &{} *\\ &{} I_r &{} &{} \vdots &{} \ddots &{} \vdots \\ &{} &{} &{} * &{} \ldots &{} *\\ \hline &{} &{} &{} &{} &{} \\ &{}\text {0}&{} &{} &{} I_s &{} \\ &{}&{} &{} &{} &{} \\ \end{array} \right) \end{aligned}$$</span></div></div><p>which by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Cabrera, C.Y., Siles, M.M., Velasco, M.V.: Evolution algebras of arbitrary dimension and their decompositions. Linear Algebra Appl. 495, 122–162 (2016)" href="/article/10.1007/s13398-023-01414-w#ref-CR4" id="ref-link-section-d2809547e8120">4</a>, Corollary 4.6], implies that <span class="mathjax-tex">${\mathcal {X}} (G)$</span> is not simple for <span class="mathjax-tex">$r,s>0$</span>.</p> </div></div></section><section data-title="Cayley evolution algebras of group algebras"><div class="c-article-section" id="Sec4-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec4"><span class="c-article-section__title-number">4 </span>Cayley evolution algebras of group algebras</h2><div class="c-article-section__content" id="Sec4-content"><p>In this section we refine the results in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s13398-023-01414-w#Sec3">3</a>. We fix a finite group <i>G</i> and we consider the group algebra over <span class="mathjax-tex">$\Bbbk $</span>, <span class="mathjax-tex">$\mathcal {A}=\Bbbk [G]$</span>. Now, we take as a basis <span class="mathjax-tex">$B=G$</span> and a function <span class="mathjax-tex">$f:G\rightarrow \Bbbk .$</span> In this setting, observe that the structure constants of <span class="mathjax-tex">$\textrm{Cay}(f)$</span> with respect to <i>G</i> can be given by <span class="mathjax-tex">$g\cdot g=\sum _{k\in G}f(g^{-1}k)k$</span>.</p><p>Our first result is a straightforward consequence of Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar9">3.4</a>.</p> <h3 class="c-article__sub-heading" id="FPar14">Proposition 4.1</h3> <p><i>G</i> acts faithfully on <span class="mathjax-tex">$\textrm{Cay}(f)$</span> as a permutation group over its natural basis by left-multiplication. That is, <span class="mathjax-tex">$G\le {\text {Aut}}(\textrm{Cay}(f))$</span>.</p> <h3 class="c-article__sub-heading" id="FPar15">Proof</h3> <p>The elements of <i>G</i> are units in <span class="mathjax-tex">$\mathcal {A}=\Bbbk [G]$</span> and since <span class="mathjax-tex">$G^2=G$</span>, then <span class="mathjax-tex">$GB=B$</span>. Hence by Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar9">3.4</a>, <span class="mathjax-tex">$\psi _g \in {\text {Aut}}\big (\textrm{Cay}(f)\big ).$</span> Moreover, for any <span class="mathjax-tex">$g,h\in G$</span>, <span class="mathjax-tex">$\psi _g\circ \psi _h=\psi _{gh}$</span>, thus the units <span class="mathjax-tex">$G\subset \mathcal {A}$</span> produce a subgroup of automorphisms which is isomorphic to <i>G</i>. <span class="mathjax-tex">$\square $</span></p> <p>We now recall the definition of Cayley graph [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="White, A.T.: Graphs, Groups and Surfaces, Volume 8 of North-Holland Mathematics Studies, 2nd edn. North-Holland Publishing Co., Amsterdam (1984)" href="/article/10.1007/s13398-023-01414-w#ref-CR17" id="ref-link-section-d2809547e8776">17</a>, Definition 4-5].</p> <h3 class="c-article__sub-heading" id="FPar16">Definition 4.2</h3> <p>Let <i>G</i> be a finite group, and <i>S</i> be a subset of <i>G</i>. The <i>Cayley graph</i> of <i>G</i> with respect to <i>S</i> is the directed graph <span class="mathjax-tex">$\textrm{Cay}(G,S)$</span> with vertex set <i>G</i> and edges (<i>g</i>, <i>gs</i>) for each <span class="mathjax-tex">$g\in G$</span> and <span class="mathjax-tex">$s\in S$</span>. Moreover, we can assign a colour <span class="mathjax-tex">$c_s$</span> to each edge (<i>g</i>, <i>gs</i>), <span class="mathjax-tex">$s\in S$</span>, where <span class="mathjax-tex">$c_s\ne c_{s'}$</span> if <span class="mathjax-tex">$s\ne s'$</span>. Hence, we obtain an <i>edge-coloured directed graph</i>, the coloured Cayley graph. In this work we will denote it by <span class="mathjax-tex">$\textrm{Cay}^{cor}(G,S)$</span>.</p> <p>The Cayley evolution algebras that we have previously introduced are closely related to Cayley graphs. By comparing Definitions <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar5">2.4</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar16">4.2</a> we immediately obtain:</p> <h3 class="c-article__sub-heading" id="FPar17">Lemma 4.3</h3> <p>Let <span class="mathjax-tex">$S={\text {supp}}(f)$</span>, and assume <span class="mathjax-tex">$f|_S$</span> is injective. Then <span class="mathjax-tex">$\Gamma \big (\textrm{Cay}(f),G\big )=\textrm{Cay}(G,S)$</span> as abstract directed graphs while <span class="mathjax-tex">$\Gamma ^w\big (\textrm{Cay}(f),G\big )=\textrm{Cay}^{cor}(G,S)$</span> as edge-coloured directed graphs.</p> <p>One of the key features of the coloured Cayley graph <span class="mathjax-tex">$\textrm{Cay}^{cor}(G,S)$</span> is that whenever <i>S</i> generates <i>G</i>, the group <i>G</i> can be represented as the full group of automorphisms of <span class="mathjax-tex">$\textrm{Cay}^{cor}(G,S)$</span>, that is <span class="mathjax-tex">$G\cong {\text {Aut}}(\textrm{Cay}^{cor}(G,S))$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="White, A.T.: Graphs, Groups and Surfaces, Volume 8 of North-Holland Mathematics Studies, 2nd edn. North-Holland Publishing Co., Amsterdam (1984)" href="/article/10.1007/s13398-023-01414-w#ref-CR17" id="ref-link-section-d2809547e9450">17</a>, Theorem 4-8]. In a similar way we prove:</p> <h3 class="c-article__sub-heading" id="FPar18">Proposition 4.4</h3> <p>Suppose that the evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra <span class="mathjax-tex">$\textrm{Cay}(f)$</span> is regular, <span class="mathjax-tex">$S={\text {supp}}(f)$</span> generates <i>G</i> and <span class="mathjax-tex">$f|_S$</span> injective. If <i>S</i> contains two elements of coprime order, then <span class="mathjax-tex">$G \cong {\text {Aut}}(\textrm{Cay}(f))$</span>.</p> <h3 class="c-article__sub-heading" id="FPar19">Proof</h3> <p>First of all, we make use of the identification <span class="mathjax-tex">$\Gamma \big (\textrm{Cay}(f),S\big )=\textrm{Cay}(G,S)$</span> as abstract directed graphs, according to Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar17">4.3</a>. Now, since <span class="mathjax-tex">$\textrm{Cay}(f)$</span> is regular, by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Elduque, A., Labra, A.: Evolution algebras, automorphisms, and graphs. Linear Multilinear Algebra 69(2), 1–12 (2019)" href="/article/10.1007/s13398-023-01414-w#ref-CR8" id="ref-link-section-d2809547e9720">8</a>, Theorem 3.2], there exists an exact sequence</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} 1\hookrightarrow {\text {Diag}}(\textrm{Cay}(G,S))\rightarrow {\text {Aut}}(\textrm{Cay}(f))\rightarrow {\text {Aut}}(\textrm{Cay}(G,S)), \end{aligned}$$</span></div></div><p>with <span class="mathjax-tex">${\text {Diag}}(\textrm{Cay}(G,S)) = \mu _N = \{ k \in {\mathbb {K}} \,; k^N =1\}$</span> for <span class="mathjax-tex">$N = 2^\mathrm{b(\textrm{Cay}(G, S))}-1 $</span> where <span class="mathjax-tex">${\text {b}}(\textrm{Cay}(G, S))$</span> is the balance of <span class="mathjax-tex">$ \textrm{Cay}(G,S)$</span> (see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Elduque, A., Labra, A.: Evolution algebras, automorphisms, and graphs. Linear Multilinear Algebra 69(2), 1–12 (2019)" href="/article/10.1007/s13398-023-01414-w#ref-CR8" id="ref-link-section-d2809547e10058">8</a>, Section 2] and [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Elduque, A., Labra, A.: Evolution algebras, automorphisms, and graphs. Linear Multilinear Algebra 69(2), 1–12 (2019)" href="/article/10.1007/s13398-023-01414-w#ref-CR8" id="ref-link-section-d2809547e10061">8</a>, Theorem 2.7]).</p> <p>We first calculate <span class="mathjax-tex">${\text {Diag}}(\textrm{Cay}(G,S))$</span>. Notice that for every <span class="mathjax-tex">$s\in S$</span>, the graph <span class="mathjax-tex">$\textrm{Cay}(G,S)$</span> has a directed cycle of length the order of the element <i>s</i>, <i>o</i>(<i>s</i>). Therefore, if <span class="mathjax-tex">$s_1,s_2\in S$</span> are elements such that <span class="mathjax-tex">$\textrm{gcd} (o(s_1),o(s_2))=1$</span>, we obtain</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} 1\le \textrm{b}(\textrm{Cay}(G,S))= \textrm{gcd}\{|\textrm{b} (\gamma )|:\, \gamma \text { cycle in} \,\textrm{Cay}(G,S) \}\le \textrm{gcd} (o(s_1),o(s_2))=1. \end{aligned}$$</span></div></div><p>Hence <span class="mathjax-tex">$N=2^{\textrm{b}(\textrm{Cay}(G,S))-1}=1$</span>, and <span class="mathjax-tex">${\text {Diag}}(\textrm{Cay}(G,S))=\mu _1=\{1\}$</span> so,</p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {Aut}}(\textrm{Cay}(f))\le {\text {Aut}}(\textrm{Cay}(G,S)). \end{aligned}$$</span></div></div><p>Therefore, elements in <span class="mathjax-tex">${\text {Aut}}(\textrm{Cay}(f))$</span> can be thought as those graph automorphisms of <span class="mathjax-tex">$\textrm{Cay}(G,S)$</span> that preserve the structure matrix <span class="mathjax-tex">$M_G(\textrm{Cay}(f))$</span> in the following sense. Let <span class="mathjax-tex">$\psi \in {\text {Aut}}(\textrm{Cay}(f))$</span> and <span class="mathjax-tex">$g \in G$</span> an arbitrary element. As <span class="mathjax-tex">$\psi $</span> preserves the product in <span class="mathjax-tex">$\textrm{Cay}(f)$</span>, for <span class="mathjax-tex">$g\cdot g=\sum _{k\in G}f(g^{-1}k)k$</span>, we have that</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \psi (g\cdot g) =\sum _{k\in G}f(g^{-1}k) \psi (k) \end{aligned}$$</span></div><div class="c-article-equation__number"> (1) </div></div><p>must coincide with <span class="mathjax-tex">$\psi (g) \cdot \psi (g) = \sum _{k\in G}f( \psi (g)^{-1}k)k$</span>. Since <span class="mathjax-tex">$\psi $</span> can be thought as a graph automorphism of <span class="mathjax-tex">$\textrm{Cay}(G,S)$</span>, <span class="mathjax-tex">$\psi $</span> is in particular a permutation of the elements in <i>G</i>, which allows us to express</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \psi (g) \cdot \psi (g) = \sum _{k \in G}f( \psi (g)^{-1}\psi (k)) \psi (k). \end{aligned}$$</span></div><div class="c-article-equation__number"> (2) </div></div><p>Comparing both (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s13398-023-01414-w#Equ1">1</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s13398-023-01414-w#Equ2">2</a>) we deduce that <span class="mathjax-tex">$ f(g^{-1}k)= f( \psi (g)^{-1}\psi (k)) $</span> for arbitrary <span class="mathjax-tex">$g, k \in G$</span>. In other words, <span class="mathjax-tex">$\psi $</span> is an automorphism of the weighted graph <span class="mathjax-tex">$\Gamma ^w\big (\textrm{Cay}(f),G\big )$</span> which, by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar17">4.3</a> coincides with <span class="mathjax-tex">$\textrm{Cay}^{cor}(G,S)$</span>. That is,</p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\text {Aut}}(\textrm{Cay}(f))\le {\text {Aut}}(\textrm{Cay}^{cor}(G,S))\cong G. \end{aligned}$$</span></div></div><p>Since <span class="mathjax-tex">$G\le {\text {Aut}}(\textrm{Cay}(f))$</span> by Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar14">4.1</a>, we conclude that <span class="mathjax-tex">$G \cong {\text {Aut}}(\textrm{Cay}(f)) $</span>. <span class="mathjax-tex">$\square $</span></p> <p>We prove now that if the field <span class="mathjax-tex">$\Bbbk $</span> is large enough compared with |<i>G</i>| (for instance, if <span class="mathjax-tex">$\Bbbk $</span> is infinite), it is then possible to construct <span class="mathjax-tex">$\textrm{Cay}(f)$</span>, a Cayley evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra associated to the function <span class="mathjax-tex">$f: G \rightarrow \Bbbk $</span>, fulfilling hypotheses from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar17">4.3</a> and Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar18">4.4</a>.</p><p>The following lemma makes the condition “large enough compared with an integer" precise. Recall that a polynomial <span class="mathjax-tex">$P\in \Bbbk [X_1,\ldots , X_m]$</span> is said to be homogeneous of degree <span class="mathjax-tex">$n\in \mathbb {N}$</span>, denoted by <span class="mathjax-tex">$\deg (P)=n$</span>, if <i>P</i> is a linear combination of monomials <span class="mathjax-tex">$X_1^{d_1}X_2^{d_2}\cdots X_m^{d_m}$</span> such that <span class="mathjax-tex">$\sum d_i=n$</span>.</p> <h3 class="c-article__sub-heading" id="FPar20">Lemma 4.5</h3> <p>Let <span class="mathjax-tex">$\Bbbk $</span> be a not necessarily finite field with multiplicative group of order <span class="mathjax-tex">$|\Bbbk ^*|$</span>, and let <i>n</i> and <i>m</i> be positive integers such that <span class="mathjax-tex">$|\Bbbk ^*|\ge 2n$</span> and <span class="mathjax-tex">$n \ge m$</span>. Then, for any homogeneous polynomial of degree <i>n</i> in <i>m</i> variables</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} P=\sum _{i=1}^m \lambda _iX_i^n + Q\in \Bbbk [X_1,\ldots , X_m], \end{aligned}$$</span></div></div><p>with <span class="mathjax-tex">$\lambda _i \ne 0$</span> for all <i>i</i>, and where every monomial in <i>Q</i> involves at least two different variables, there exists a choice of non-zero pairwise distinct values <span class="mathjax-tex">$k_1,\ldots , k_m\in \Bbbk ^*$</span> such that <span class="mathjax-tex">$P(X_i=k_i: i=1,\ldots , m)\ne 0$</span>.</p> <h3 class="c-article__sub-heading" id="FPar21">Proof</h3> <p>Fix <span class="mathjax-tex">$n \ge m$</span> and <span class="mathjax-tex">$P\in \Bbbk [X_1,\ldots , X_m]$</span> as in the hypotheses. Since <span class="mathjax-tex">$|\Bbbk ^*| \ge m$</span>, we can choose <span class="mathjax-tex">$k_1,\ldots , k_{m-1}\in \Bbbk ^*$</span> pairwise distinct values. Hence, the polynomial <span class="mathjax-tex">$\widetilde{P}(X_m):=P(X_i=k_i: i<m)$</span> is a non-trivial polynomial of degree <i>n</i>, and therefore the equation <span class="mathjax-tex">$\widetilde{P}(X_m)=0$</span> admits a most <i>n</i> different non-zero solutions. That is, there exist at most <span class="mathjax-tex">$\kappa _1,\ldots ,\kappa _n\in \Bbbk ^*$</span> such that <span class="mathjax-tex">$\widetilde{P}(X_m=\kappa _r)=0$</span>, for <span class="mathjax-tex">$r=1, \ldots , n$</span>. Since <span class="mathjax-tex">$|\Bbbk ^*|\ge 2n> m-1+n$</span>, there exists a non-zero value <span class="mathjax-tex">$k_m\in \Bbbk ^*$</span> such that <span class="mathjax-tex">$k_m\ne k_j$</span> for <span class="mathjax-tex">$j=1,\ldots ,m-1$</span> and <span class="mathjax-tex">$k_m\ne \kappa _r$</span> for <span class="mathjax-tex">$r=1,\ldots , n$</span>. Then, <span class="mathjax-tex">$k_1,\ldots , k_m\in \Bbbk ^*$</span> are non-zero pairwise distinct values and <span class="mathjax-tex">$P(X_i=k_i: i=1,\ldots , m)\ne 0$</span>. <span class="mathjax-tex">$\square $</span></p> <h3 class="c-article__sub-heading" id="FPar22">Theorem 4.6</h3> <p>Let <span class="mathjax-tex">$\Bbbk $</span> be a not necessarily finite field with multiplicative group of order <span class="mathjax-tex">$|\Bbbk ^*|$</span> and let <i>G</i> be a finite group. If <span class="mathjax-tex">$|\Bbbk ^*|\ge 2 |G|$</span>, then for any non-empty <span class="mathjax-tex">$S\subset G$</span> there exists a function <span class="mathjax-tex">$f :G\rightarrow \Bbbk $</span> such that: </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p><span class="mathjax-tex">$S={\text {supp}}(f)$</span></p> </li> <li> <span class="u-custom-list-number">(2)</span> <p><span class="mathjax-tex">$f|_S$</span> is injective</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p><span class="mathjax-tex">$\textrm{Cay}(f)$</span> is regular.</p> </li> </ol> <h3 class="c-article__sub-heading" id="FPar23">Proof</h3> <p>For any function <span class="mathjax-tex">$f:G\rightarrow \Bbbk $</span>, if we denote by <span class="mathjax-tex">$X_g$</span> the variable such that <span class="mathjax-tex">$X_g=f(g)$</span>, the structure coefficients of <span class="mathjax-tex">$\textrm{Cay}(f)$</span> can be described by</p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} g\cdot g=\sum _{h\in G}f(h)gh=\sum _{k\in G}f(g^{-1}k)k=\sum _{k\in G}X_{g^{-1}k}k, \end{aligned}$$</span></div></div><p>and therefore, the structure matrix of <span class="mathjax-tex">$\textrm{Cay}(f)$</span> becomes</p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} M_B(\textrm{Cay}(f))=(X_{g^{-1}k})_{g,k\in G}\in M_{|G|\times |G|}(\Bbbk ), \end{aligned}$$</span></div></div><p>whose determinant is a homogeneous polynomial <span class="mathjax-tex">$P(X_g:\,g\in G)$</span> of degree |<i>G</i>|, that we now describe. Given <span class="mathjax-tex">$k\in G$</span>, let <span class="mathjax-tex">$P_k:=P(X_g=0: g\ne k)\in \Bbbk [X_k],$</span> thus <span class="mathjax-tex">$P=\sum _{k\in G} P_k + Q$</span> where <i>Q</i> is a homogeneous polynomial in <span class="mathjax-tex">$\Bbbk [X_g: g\in G]$</span> such that every monomial in <i>Q</i> involves at least two different variables. Since <i>P</i> is homogeneous, <span class="mathjax-tex">$P_k(X_k)=\lambda _k X_k^{|G|}$</span> for some <span class="mathjax-tex">$\lambda _k\in \Bbbk $</span>. Following the notation of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Harary, F.: The determinant of the adjacency matrix of a graph. Siam Rev. 4(3), 202–210 (1962)" href="/article/10.1007/s13398-023-01414-w#ref-CR10" id="ref-link-section-d2809547e14508">10</a>], we can think of the polynomials <span class="mathjax-tex">$P_k$</span> as</p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} P_k(X_k)=\det \Big (\textrm{A}\big (\textrm{Cay}(G,\{k\}), X_k\big )\Big ) \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\textrm{A}\big (\textrm{Cay}(G,\{k\}), X_k\big )$</span> denotes the variable adjacency matrix of the Cayley graph <span class="mathjax-tex">$\textrm{Cay}(G,\{k\})$</span>. And, according to [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Harary, F.: The determinant of the adjacency matrix of a graph. Siam Rev. 4(3), 202–210 (1962)" href="/article/10.1007/s13398-023-01414-w#ref-CR10" id="ref-link-section-d2809547e14762">10</a>, Theorem 2], as <span class="mathjax-tex">$\textrm{Cay}(G,\{k\})$</span> is a directed graph consisting of a total of <span class="mathjax-tex">$\frac{|G|}{o(k)}$</span> disjoint directed cycles of length <i>o</i>(<i>k</i>), which are also the strong components of <span class="mathjax-tex">$\textrm{Cay}(G,\{k\})$</span>, we obtain that</p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} P_k(X_k)= \Big ( \det \big ( \textrm{A}\big (o(k)\text {-cycle}, X_k \big ) \big ) \Big )^{\frac{|G|}{o(k)}}. \end{aligned}$$</span></div></div><p>Since the variable adjacency matrix of a directed <i>o</i>(<i>k</i>)-cycle is of the form</p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left( \begin{array}{ccccc} 0 &{} X_{k} &{} 0 &{} \ldots &{} 0\\ 0 &{} 0 &{} X_{k} &{} \ldots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \ldots &{} X_{k}\\ X_{k} &{} 0 &{} 0 &{} \ldots &{} 0\\ \end{array} \right) \in M_{o(k) \times o(k)} (\Bbbk ) \end{aligned}$$</span></div></div><p>we conclude that</p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} P_k(X_k)=\Big ((-1)^{o(k)+1}X_{k}^{o(k)}\Big )^{\frac{|G|}{o(k)}}= (-1)^{\frac{(o(k)+1)|G|}{o(k)}} X_k^{|G|}. \end{aligned}$$</span></div></div><p>We proceed now to construct, for a given <span class="mathjax-tex">$S \subset G$</span>, the desired function <span class="mathjax-tex">$f:G\rightarrow \Bbbk $</span>. Let <span class="mathjax-tex">$P_S:=P(X_g=0: g\not \in S)\in \Bbbk [X_s:s\in S]$</span> which is a non-trivial homogeneous polynomial of degree |<i>G</i>| on |<i>S</i>| variables. As <span class="mathjax-tex">$|\Bbbk ^*| \ge 2n$</span>, Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar20">4.5</a> ensures the existence of non-zero pairwise distinct values <span class="mathjax-tex">$k_s\in \Bbbk ^*$</span>, <span class="mathjax-tex">$s\in S$</span>, satisfying that <span class="mathjax-tex">$P_S(X_s=k_s: s\in S)\ne 0$</span>. With that in mind, let <span class="mathjax-tex">$f:G\rightarrow \Bbbk $</span> be defined by</p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} f(g)={\left\{ \begin{array}{ll} k_g, &{} \text {if } g\in S \\ 0, &{} \text {otherwise} \end{array}\right. } \end{aligned}$$</span></div></div><p>Notice that since all the values <span class="mathjax-tex">$k_s\in \Bbbk ^*$</span>, for <span class="mathjax-tex">$s\in S$</span>, are non-zero and pairwise distinct, <span class="mathjax-tex">$S={\text {supp}}(f)$</span> and <span class="mathjax-tex">$f|_S$</span> is injective. Finally,</p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} \det \big (M_B(\textrm{Cay}(f))\big )&= P(X_g=f(g):g\in G)\\&= P_S(X_s=k_s:s\in S) \\&\ne 0 \end{aligned} \end{aligned}$$</span></div></div><p>hence <span class="mathjax-tex">$\textrm{Cay}(f)$</span> is regular. <span class="mathjax-tex">$\square $</span></p> <p>Combining the results previously obtained, we give the proof of our main result.</p> <h3 class="c-article__sub-heading" id="FPar24">Proof of Theorem A</h3> <p>Let <i>S</i> be a set of generators of <i>G</i> containing coprime order elements. Observe that <i>S</i> is also a set of generators of <span class="mathjax-tex">$\Bbbk [G] $</span> as an algebra, and such <i>S</i> always exists, it suffices to consider a generating set <i>S</i> containing <span class="mathjax-tex">$1\in S$</span>, for example <span class="mathjax-tex">$S = G$</span>. Considering the function <span class="mathjax-tex">$f:G\rightarrow \Bbbk $</span> given in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar22">4.6</a>, <span class="mathjax-tex">$\textrm{Cay}(f)$</span> is regular, <span class="mathjax-tex">$S={\text {supp}}(f)$</span> and <span class="mathjax-tex">$f|_S$</span> is injective. Moreover, <i>S</i> contains coprime order elements, so by Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar18">4.4</a> we conclude that <span class="mathjax-tex">${\text {Aut}}(\textrm{Cay}(f))\cong G$</span>. Moreover, since <i>S</i> generates <span class="mathjax-tex">$\Bbbk [G]$</span> and <span class="mathjax-tex">$\textrm{Cay}(f)$</span> is regular, by Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar11">3.5</a> we obtain that <span class="mathjax-tex">$\textrm{Cay}(f)$</span> is simple.</p> <p>We now check that the simple evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra <span class="mathjax-tex">$X =\textrm{Cay}(f) $</span> above is indeed absolutely simple, and <span class="mathjax-tex">${\text {Aut}}(X_{{\mathbb {F}}})\cong G$</span> for every field extension <span class="mathjax-tex">$\mathbb {F}/\Bbbk $</span>. In fact, since the function <span class="mathjax-tex">$f_{{\mathbb {F}}}:G\rightarrow {{\mathbb {F}}}$</span> induced by <i>f</i> also satisfies conclusions (1), (2), and (3) in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar22">4.6</a>, <span class="mathjax-tex">${\text {Aut}}(\textrm{Cay}(f_{{\mathbb {F}}}))\cong G$</span>, and <span class="mathjax-tex">$\textrm{Cay}(f_{{\mathbb {F}}})$</span> is simple. Finally, since <span class="mathjax-tex">${{\mathbb {F}}}[G]=\Bbbk [G]\otimes _\Bbbk {{\mathbb {F}}}$</span> then <span class="mathjax-tex">$\textrm{Cay}(f_{{\mathbb {F}}})=\textrm{Cay}(f)\otimes _\Bbbk {{\mathbb {F}}}=X_{{\mathbb {F}}}$</span>, thus <span class="mathjax-tex">$X_{{\mathbb {F}}}$</span> is simple and <span class="mathjax-tex">${\text {Aut}}(X_{{\mathbb {F}}})\cong G$</span>.</p> <p> <span class="mathjax-tex">$\square $</span> </p> <p>We end this paper with some remarks on the non-uniqueness of the simple evolution algebras realizing finite groups.</p> <h3 class="c-article__sub-heading" id="FPar25">Remark 4.7</h3> <p>Our construction of the simple evolution <span class="mathjax-tex">$\Bbbk $</span>-algebra <span class="mathjax-tex">$\textrm{Cay}(f) $</span> such that <span class="mathjax-tex">${\text {Aut}}(\textrm{Cay}(f))\cong G$</span>, for a fixed finite group <i>G</i> involves several choices: first, a generator set <span class="mathjax-tex">$S\subset G$</span> containing coprime order elements, which is not unique if <span class="mathjax-tex">$|G|>1$</span>. Secondly, a function <span class="mathjax-tex">$f:G\rightarrow \Bbbk $</span> such that <span class="mathjax-tex">$\textrm{Cay}(f)$</span> is regular, as in the proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar22">4.6</a>, which is not is unique either since any non-zero scalar multiple of <span class="mathjax-tex">$f:G\rightarrow \Bbbk $</span> will also work. An easy argument on the edge-coloured directed graphs <span class="mathjax-tex">$\Gamma ^w\big (\textrm{Cay}(f),B\big )=\textrm{Cay}^{cor}(G,S)$</span> illustrates that both choices lead to non-isomorphic evolution <span class="mathjax-tex">$\Bbbk $</span>-algebras with the same group of automorphism <i>G</i>.</p> <h3 class="c-article__sub-heading" id="FPar26">Remark 4.8</h3> <p>For any evolution algebra <span class="mathjax-tex">$\textrm{Cay}(f)$</span> built upon the group algebra <span class="mathjax-tex">$\mathcal {A}=\Bbbk [G]$</span> with fixed natural basis <span class="mathjax-tex">$B=G$</span>, and any given <span class="mathjax-tex">$f:B\rightarrow \Bbbk $</span>, the following holds</p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} G\le {\text {Aut}}(\textrm{Cay}(f))\le \textrm{GL}(\Bbbk , |G|) \end{aligned}$$</span></div></div><p>since <span class="mathjax-tex">$B=G$</span> is a group of units basis in <span class="mathjax-tex">$\mathcal {A}=\Bbbk [G]$</span> (see Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar14">4.1</a>). The lower bound <span class="mathjax-tex">$G= {\text {Aut}}(\textrm{Cay}(f))$</span> is obtained when <i>f</i> is as in the proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar1">A</a> (see also Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar22">4.6</a>) while the upper bound <span class="mathjax-tex">${\text {Aut}}(\textrm{Cay}(f))=\textrm{GL}(\Bbbk , |G|)$</span> is obtained when <span class="mathjax-tex">$f= 0$</span>. Although it is not the purpose of this paper, we could raise the question of which intermediate groups <i>H</i>,</p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} G\le H\le \textrm{GL}(\Bbbk , |G|), \end{aligned}$$</span></div></div><p>can be realized as automorphisms of simple Cayley evolution algebras. That is, <span class="mathjax-tex">${\text {Aut}}(\textrm{Cay}(f))= H$</span> for some function <span class="mathjax-tex">$f:G\rightarrow \Bbbk $</span>.</p> <p>We now highlight that not only <span class="mathjax-tex">${\text {Aut}}(\textrm{Cay}(f))$</span> depends on how <i>good</i> or <i>bad</i> the function <span class="mathjax-tex">$f: G \rightarrow \Bbbk $</span> is, as Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar26">4.8</a> illustrates, but it also depends on the group of units of <span class="mathjax-tex">$\mathcal {A}=\Bbbk [G]$</span>. And in fact, since the isomorphism problem for group rings [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Sandling, R.: The isomorphism problem for group rings: A survey. In: Reiner, I., Roggenkamp, K.W. (eds.) Orders and Their Applications, pp. 256–288. Springer, Berlin (1985)" href="/article/10.1007/s13398-023-01414-w#ref-CR14" id="ref-link-section-d2809547e18343">14</a>, Problem 1.1] has a negative answer in general [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Hertweck, M.: A counterexample to the isomorphism problem for integral group rings. Ann. Math. (2) 154(1), 115–138 (2001)" href="/article/10.1007/s13398-023-01414-w#ref-CR11" id="ref-link-section-d2809547e18346">11</a>], the group of units basis is not unique in a group algebra. The following example shows that the choice of the group of units basis as natural basis for <span class="mathjax-tex">$\textrm{Cay}(f)$</span> might lead to non-isomorphic Cayley evolution algebras with non-isomorphic group of automorphisms:</p> <h3 class="c-article__sub-heading" id="FPar27">Example 4.9</h3> <p>Let <span class="mathjax-tex">$G_1$</span> and <span class="mathjax-tex">$G_2$</span> be the finite groups of order <span class="mathjax-tex">$n=2^{21}97^{28}$</span> denoted by respectively <i>X</i> and <i>Y</i> in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Hertweck, M.: A counterexample to the isomorphism problem for integral group rings. Ann. Math. (2) 154(1), 115–138 (2001)" href="/article/10.1007/s13398-023-01414-w#ref-CR11" id="ref-link-section-d2809547e18471">11</a>, Theorem B]. Let <span class="mathjax-tex">$\Bbbk $</span> be a field (not necessarily finite) with multiplicative group of order <span class="mathjax-tex">$|\Bbbk ^*| \ge 2 \cdot 2^{21}97^{28}$</span> and let <span class="mathjax-tex">$\mathcal {A}= \Bbbk [G_1]$</span> be the group <span class="mathjax-tex">$\Bbbk $</span>-algebra. By [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Hertweck, M.: A counterexample to the isomorphism problem for integral group rings. Ann. Math. (2) 154(1), 115–138 (2001)" href="/article/10.1007/s13398-023-01414-w#ref-CR11" id="ref-link-section-d2809547e18608">11</a>, Theorem B], although <span class="mathjax-tex">$G_1$</span> and <span class="mathjax-tex">$G_2$</span> are non-isomorphic, they can both be the group of units basis of <span class="mathjax-tex">$\mathcal {A}$</span>. Hence, following the proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar1">A</a>, there exist functions <span class="mathjax-tex">$f_1:G_1\rightarrow \Bbbk $</span>, and <span class="mathjax-tex">$f_2:G_2\rightarrow \Bbbk $</span>, such that <span class="mathjax-tex">$\textrm{Cay}(f_1)$</span> and <span class="mathjax-tex">$\textrm{Cay}(f_2)$</span> are simple evolution <span class="mathjax-tex">$\Bbbk $</span>-algebras with <span class="mathjax-tex">${\text {Aut}}(\textrm{Cay}(f_1))\cong G_1 \not \cong G_2 \cong {\text {Aut}}(\textrm{Cay}(f_2))$</span>, and so <span class="mathjax-tex">$\textrm{Cay}(f_1) \not \cong \textrm{Cay}(f_2)$</span>.</p> <p>Finally, it is also natural to ask whether the function <span class="mathjax-tex">$f:G\rightarrow \Bbbk $</span> defining <span class="mathjax-tex">$\textrm{Cay}(f)$</span> can be chosen to be a significant one, for example a character, or more generally, a class function, that is <span class="mathjax-tex">$f(g^{-1}kg)=f(k)$</span> for all <span class="mathjax-tex">$g,k\in G$</span>. We prove the following:</p> <h3 class="c-article__sub-heading" id="FPar28">Proposition 4.10</h3> <p>Let <i>G</i> be a finite group, <span class="mathjax-tex">$\mathcal {A}=\Bbbk [G]$</span>, and <span class="mathjax-tex">$f:G\rightarrow \Bbbk $</span> be a class function. Then <span class="mathjax-tex">${\text {Aut}}(\textrm{Cay}(f))$</span> contains two subgroups <span class="mathjax-tex">$K_i$</span>, <span class="mathjax-tex">$i=1,2$</span>, such that <span class="mathjax-tex">$K_1\cong G$</span>, <span class="mathjax-tex">$K_2\cong G/Z(G)$</span> and <span class="mathjax-tex">$K_1\cap K_2=\{1\}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar29">Proof</h3> <p>By Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar14">4.1</a>, there exists <span class="mathjax-tex">$K_1\cong G\le {\text {Aut}}(\textrm{Cay}(f))$</span>. Moreover, since the natural basis <span class="mathjax-tex">$B=G$</span> of <span class="mathjax-tex">$\textrm{Cay}(f)$</span> consists of units in <span class="mathjax-tex">$\mathcal {A}=\Bbbk [G]$</span>, Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar9">3.4</a> ensures that <span class="mathjax-tex">$K_1$</span> acts freely on the natural basis of <span class="mathjax-tex">$\textrm{Cay}(f)$</span>.</p> <p>We now consider the following morphism <span class="mathjax-tex">$\rho :G\rightarrow {\text {Aut}}(\textrm{Cay}(f))$</span>: for any <span class="mathjax-tex">$h\in G$</span>, let <span class="mathjax-tex">$\rho (h)$</span> be the linear automorphism of <span class="mathjax-tex">$\textrm{Cay}(f)=\Bbbk [G]$</span> given by inner conjugation in <i>G</i>, that is, <span class="mathjax-tex">$\rho (h)(g)=hgh^{-1}$</span>. Observe that <span class="mathjax-tex">$\rho (h)$</span> maps the natural basis <span class="mathjax-tex">$B=G$</span> of <span class="mathjax-tex">$\textrm{Cay}(f)$</span> to itself, hence if <span class="mathjax-tex">$g \ne g'$</span> thus <span class="mathjax-tex">$g\cdot g'=0$</span>, then <span class="mathjax-tex">$\rho (h)(g) \ne \rho (h)(g')$</span> and <span class="mathjax-tex">$\rho (h)(g)\cdot \rho (h)(g')=0$</span>. Moreover</p><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{aligned} \rho (h)(g\cdot g)&=\rho (h)\left( \sum _{k\in G}f(g^{-1}k)k\right) \\&=\sum _{k\in G}f(g^{-1}k)\rho (h)(k)\\&=\sum _{k\in G}f(g^{-1}k) hkh^{-1}\\&=\sum _{k\in G}f(hg^{-1}kh^{-1}) hkh^{-1}\text { (since { f} is a class function)}\\&=\sum _{k\in G}f(hg^{-1}h^{-1}hkh^{-1}) hkh^{-1}\\&=\sum _{k\in G}f(hg^{-1}h^{-1}k) k\\&=\sum _{k\in G}f(\rho (h)(g)^{-1}k) k\\&= \rho (h)(g)\cdot \rho (h)(g). \end{aligned} \end{aligned}$$</span></div></div><p>Therefore, <span class="mathjax-tex">$\rho (h)$</span> is an actual automorphism of <span class="mathjax-tex">$\textrm{Cay}(f)$</span> for every <span class="mathjax-tex">$h\in G$</span> and by construction <span class="mathjax-tex">$\rho (h)=\textrm{Id}$</span> if and only if <span class="mathjax-tex">$h\in Z(G)$</span>. Hence <span class="mathjax-tex">$\ker (\rho )=Z(G)$</span> and <span class="mathjax-tex">$K_2={\text {Im}}(\rho )\cong G/Z(G)$</span>.</p> <p>Finally, notice that every <span class="mathjax-tex">$\rho (h)\in K_2$</span> fixes, at least, the element <i>h</i> in <i>B</i>, while every non-trivial element in <span class="mathjax-tex">$K_1$</span> acts freely on <i>B</i>. Therefore <span class="mathjax-tex">$K_1\cap K_2=\{1\}$</span>. <span class="mathjax-tex">$\square $</span></p> <p>We obtain the following corollary:</p> <h3 class="c-article__sub-heading" id="FPar30">Corollary 4.11</h3> <p>Let <span class="mathjax-tex">$\mathcal {A}=\Bbbk [G]$</span>, and <span class="mathjax-tex">$f:G\rightarrow \Bbbk $</span> be a class function satisfying that <span class="mathjax-tex">${\text {Aut}}(\textrm{Cay}(f))\cong G$</span>. Then <i>G</i> is abelian.</p> <h3 class="c-article__sub-heading" id="FPar31">Proof</h3> <p>According to Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s13398-023-01414-w#FPar28">4.10</a>, as <span class="mathjax-tex">${\text {Aut}}(\textrm{Cay}(f))\cong G$</span>, the subgroup <i>G</i>/<i>Z</i>(<i>G</i>) must be trivial.</p> <p> <span class="mathjax-tex">$\square $</span> </p> </div></div></section> </div> <div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">Ahmed, H., Bekbaev, U., Rakhimov, I.: On classification of <span class="mathjax-tex">$2$</span>-dimensional evolution algebras and its applications. 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This work was partially supported by MCIN/AEI/10.13039/501100011033 [PID2020-115155GB-I00 and TED2021-131201B-I00 to C.C., PID2020-118452GB-I00 to V.M., PID2019-104236GB-I00 to A.T, and PID2020–118753GB-I00 to A.V.], and by Junta de Andalucía [UMA18-FEDERJA-119, FQM-336 to A.T., and PROYEXCEL-00827, FQM-213 to A.V.].</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">CITIC, CITMAGA, Departamento de Ciencias de la Computación y Tecnologías de la Información, Universidade da Coruña, 15071-A, Coruna, Spain</p><p class="c-article-author-affiliation__authors-list">C. Costoya</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, 29071, Malaga, Spain</p><p class="c-article-author-affiliation__authors-list">V. Muñoz & A. Viruel</p></li><li id="Aff3"><p class="c-article-author-affiliation__address">Departamento de Matemática Aplicada, Universidad de Málaga, 29071, Malaga, Spain</p><p class="c-article-author-affiliation__authors-list">A. Tocino</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-C_-Costoya-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">C. 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Costoya, C., Muñoz, V., Tocino, A. <i>et al.</i> Automorphism groups of Cayley evolution algebras. <i>Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.</i> <b>117</b>, 82 (2023). https://doi.org/10.1007/s13398-023-01414-w</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s13398-023-01414-w?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2022-07-08">08 July 2022</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Accepted<span 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initiative </p></div></div><h3 class="c-article__sub-heading">Keywords</h3><ul class="c-article-subject-list"><li class="c-article-subject-list__subject"><span><a href="/search?query=Evolution%20algebra&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Evolution algebra</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Finite%20group&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Finite group</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Automorphism%20group&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Automorphism group</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Graph&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Graph</a></span></li></ul><h3 class="c-article__sub-heading">Mathematics Subject Classification</h3><ul class="c-article-subject-list"><li class="c-article-subject-list__subject"><span><a href="/search?query=05C25&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">05C25</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=17A36&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">17A36</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=17D99&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">17D99</a></span></li></ul><div data-component="article-info-list"></div></div></div></div></div></section> </div> </main> <div class="c-article-sidebar u-text-sm u-hide-print l-with-sidebar__sidebar" id="sidebar" 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