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href="https://de.wikipedia.org/wiki/Frattinigruppe" title="Frattinigruppe – German" lang="de" hreflang="de" data-title="Frattinigruppe" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Sous-groupe_de_Frattini" title="Sous-groupe de Frattini – French" lang="fr" hreflang="fr" data-title="Sous-groupe de Frattini" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%84%EB%9D%BC%ED%8B%B0%EB%8B%88_%EB%B6%80%EB%B6%84%EA%B5%B0" title="프라티니 부분군 – Korean" lang="ko" hreflang="ko" data-title="프라티니 부분군" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Subgrup_Frattini" title="Subgrup Frattini – Indonesian" lang="id" hreflang="id" data-title="Subgrup Frattini" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sottogruppo_di_Frattini" title="Sottogruppo di Frattini – Italian" lang="it" hreflang="it" data-title="Sottogruppo di Frattini" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%AA-%D7%97%D7%91%D7%95%D7%A8%D7%AA_%D7%A4%D7%A8%D7%98%D7%99%D7%A0%D7%99" title="תת-חבורת פרטיני – Hebrew" lang="he" hreflang="he" data-title="תת-חבורת פרטיני" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Frattini-r%C3%A9szcsoport" title="Frattini-részcsoport – Hungarian" lang="hu" hreflang="hu" data-title="Frattini-részcsoport" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%83%A9%E3%83%83%E3%83%86%E3%82%A3%E3%83%BC%E3%83%8B%E9%83%A8%E5%88%86%E7%BE%A4" title="フラッティーニ部分群 – Japanese" lang="ja" hreflang="ja" data-title="フラッティーニ部分群" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Podgrupa_Frattiniego" title="Podgrupa Frattiniego – Polish" lang="pl" hreflang="pl" data-title="Podgrupa Frattiniego" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%96%D0%B4%D0%B3%D1%80%D1%83%D0%BF%D0%B0_%D0%A4%D1%80%D0%B0%D1%82%D1%82%D1%96%D0%BD%D1%96" title="Підгрупа Фраттіні – Ukrainian" lang="uk" hreflang="uk" data-title="Підгрупа Фраттіні" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Nh%C3%B3m_con_Frattini" title="Nhóm con Frattini – Vietnamese" lang="vi" hreflang="vi" data-title="Nhóm con Frattini" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li> </ul> <div 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/File:Dih4_subgroups_(cycle_graphs).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Dih4_subgroups_%28cycle_graphs%29.svg/360px-Dih4_subgroups_%28cycle_graphs%29.svg.png" decoding="async" width="360" height="299" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Dih4_subgroups_%28cycle_graphs%29.svg/540px-Dih4_subgroups_%28cycle_graphs%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Dih4_subgroups_%28cycle_graphs%29.svg/720px-Dih4_subgroups_%28cycle_graphs%29.svg.png 2x" data-file-width="1009" data-file-height="839" /></a><figcaption><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> of the <a href="/wiki/Lattice_of_subgroups" title="Lattice of subgroups">lattice of subgroups</a> of the <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral group</a> <a href="/wiki/Dihedral_group_of_order_8" class="mw-redirect" title="Dihedral group of order 8">Dih<sub>4</sub>.</a> In the second row are the maximal subgroups; their intersection (the <b>Frattini subgroup</b>) is the central element in the third row. So Dih<sub>4</sub> has only one non-generating element beyond <i>e</i>.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, particularly in <a href="/wiki/Group_theory" title="Group theory">group theory</a>, the <b>Frattini subgroup</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97079a61070b889109c86f1e0873652ddb081c3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.314ex; height:2.843ex;" alt="{\displaystyle \Phi (G)}"></span> of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> <span class="texhtml mvar" style="font-style:italic;">G</span> is the <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of all <a href="/wiki/Maximal_subgroup" title="Maximal subgroup">maximal subgroups</a> of <span class="texhtml mvar" style="font-style:italic;">G</span>. For the case that <span class="texhtml mvar" style="font-style:italic;">G</span> has no maximal subgroups, for example the <a href="/wiki/Trivial_group" title="Trivial group">trivial group</a> {<i>e</i>} or a <a href="/wiki/Pr%C3%BCfer_group" title="Prüfer group">Prüfer group</a>, it is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (G)=G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (G)=G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf076d80926aecfee94e84ca77f8d10150f76d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.239ex; height:2.843ex;" alt="{\displaystyle \Phi (G)=G}"></span>. It is analogous to the <a href="/wiki/Jacobson_radical" title="Jacobson radical">Jacobson radical</a> in the theory of <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, and intuitively can be thought of as the subgroup of "small elements" (see the "non-generator" characterization below). It is named after <a href="/wiki/Giovanni_Frattini" title="Giovanni Frattini">Giovanni Frattini</a>, who defined the concept in a paper published in 1885.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Some_facts">Some facts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frattini_subgroup&action=edit&section=1" title="Edit section: Some facts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97079a61070b889109c86f1e0873652ddb081c3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.314ex; height:2.843ex;" alt="{\displaystyle \Phi (G)}"></span> is equal to the set of all <b>non-generators</b> or <b>non-generating elements</b> of <span class="texhtml mvar" style="font-style:italic;">G</span>. A non-generating element of <span class="texhtml mvar" style="font-style:italic;">G</span> is an element that can always be removed from a <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generating set</a>; that is, an element <i>a</i> of <span class="texhtml mvar" style="font-style:italic;">G</span> such that whenever <span class="texhtml mvar" style="font-style:italic;">X</span> is a generating set of <span class="texhtml mvar" style="font-style:italic;">G</span> containing <i>a</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\setminus \{a\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\setminus \{a\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0484fa4afbcb5e0cdc4d6cade7d85739d940cfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.729ex; height:2.843ex;" alt="{\displaystyle X\setminus \{a\}}"></span> is also a generating set of <span class="texhtml mvar" style="font-style:italic;">G</span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97079a61070b889109c86f1e0873652ddb081c3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.314ex; height:2.843ex;" alt="{\displaystyle \Phi (G)}"></span> is always a <a href="/wiki/Characteristic_subgroup" title="Characteristic subgroup">characteristic subgroup</a> of <span class="texhtml mvar" style="font-style:italic;">G</span>; in particular, it is always a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of <span class="texhtml mvar" style="font-style:italic;">G</span>.</li> <li>If <span class="texhtml mvar" style="font-style:italic;">G</span> is finite, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97079a61070b889109c86f1e0873652ddb081c3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.314ex; height:2.843ex;" alt="{\displaystyle \Phi (G)}"></span> is <a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a>.</li> <li>If <span class="texhtml mvar" style="font-style:italic;">G</span> is a finite <a href="/wiki/P-group" title="P-group"><i>p</i>-group</a>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (G)=G^{p}[G,G]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">[</mo> <mi>G</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (G)=G^{p}[G,G]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0889fe5ab2e74d82eefbb7332e0abe1879d5aab8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.279ex; height:2.843ex;" alt="{\displaystyle \Phi (G)=G^{p}[G,G]}"></span>. Thus the Frattini subgroup is the smallest (with respect to inclusion) <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> <i>N</i> such that the <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G/N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab52bff253c690c4e0d473400ab8c365ea019298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/N}"></span> is an <a href="/wiki/Elementary_abelian_group" title="Elementary abelian group">elementary abelian group</a>, i.e., <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> to a <a href="/wiki/Direct_sum_of_abelian_groups" class="mw-redirect" title="Direct sum of abelian groups">direct sum</a> of <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic groups</a> of <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> <i>p</i>. Moreover, if the quotient group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/\Phi (G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G/\Phi (G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adad421dba00dd858c24bb0d1de3e9e9a110352c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.303ex; height:2.843ex;" alt="{\displaystyle G/\Phi (G)}"></span> (also called the <i>Frattini quotient</i> of <span class="texhtml mvar" style="font-style:italic;">G</span>) has order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e017c102135ab13bdf501dc1c1b5fd1840a97822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.348ex; height:3.009ex;" alt="{\displaystyle p^{k}}"></span>, then <i>k</i> is the smallest number of generators for <span class="texhtml mvar" style="font-style:italic;">G</span> (that is, the smallest cardinality of a generating set for <span class="texhtml mvar" style="font-style:italic;">G</span>). In particular a finite <i>p</i>-group is cyclic <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> its Frattini quotient is cyclic (of order <i>p</i>). A finite <i>p</i>-group is elementary abelian if and only if its Frattini subgroup is the <a href="/wiki/Trivial_group" title="Trivial group">trivial group</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (G)=\{e\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>e</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (G)=\{e\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb0cbdb190a30c82c1c1d91d6a9e6ec8d0370c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.821ex; height:2.843ex;" alt="{\displaystyle \Phi (G)=\{e\}}"></span>.</li> <li>If <span class="texhtml mvar" style="font-style:italic;">H</span> and <span class="texhtml mvar" style="font-style:italic;">K</span> are finite, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (H\times K)=\Phi (H)\times \Phi (K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo>×</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (H\times K)=\Phi (H)\times \Phi (K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce756600fe89fd24cfed01192d0cd93a1357f1a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.5ex; height:2.843ex;" alt="{\displaystyle \Phi (H\times K)=\Phi (H)\times \Phi (K)}"></span>.</li></ul> <p>An example of a group with nontrivial Frattini subgroup is the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> <span class="texhtml mvar" style="font-style:italic;">G</span> of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef685027b97072ee63a8c738f395cd40f63767e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:3.009ex;" alt="{\displaystyle p^{2}}"></span>, where <i>p</i> is prime, generated by <i>a</i>, say; here, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (G)=\left\langle a^{p}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>⟨</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (G)=\left\langle a^{p}\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62df588cf8ff2ae78aa3d5976e5d7e5676ffee2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.511ex; height:2.843ex;" alt="{\displaystyle \Phi (G)=\left\langle a^{p}\right\rangle }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frattini_subgroup&action=edit&section=2" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fitting_subgroup" title="Fitting subgroup">Fitting subgroup</a></li> <li><a href="/wiki/Frattini%27s_argument" title="Frattini's argument">Frattini's argument</a></li> <li><a href="/wiki/Socle_(mathematics)" title="Socle (mathematics)">Socle</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frattini_subgroup&action=edit&section=3" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFFrattini1885" class="citation journal cs1"><a href="/wiki/Giovanni_Frattini" title="Giovanni Frattini">Frattini, Giovanni</a> (1885). <a rel="nofollow" class="external text" href="http://www.advgrouptheory.com/GTArchivum/Frattini/FrattiniPaper1885Transl.pdf">"Intorno alla generazione dei gruppi di operazioni"</a> <span class="cs1-format">(PDF)</span>. <i>Accademia dei Lincei, Rendiconti</i>. (4). <b>I</b>: 281–285, 455–457. <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:17.0097.01">17.0097.01</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Accademia+dei+Lincei%2C+Rendiconti&rft.atitle=Intorno+alla+generazione+dei+gruppi+di+operazioni&rft.volume=I&rft.pages=281-285%2C+455-457&rft.date=1885&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A17.0097.01%23id-name%3DJFM&rft.aulast=Frattini&rft.aufirst=Giovanni&rft_id=http%3A%2F%2Fwww.advgrouptheory.com%2FGTArchivum%2FFrattini%2FFrattiniPaper1885Transl.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrattini+subgroup" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall1959" class="citation book cs1"><a href="/wiki/Marshall_Hall_(mathematician)" title="Marshall Hall (mathematician)">Hall, Marshall</a> (1959). <i>The Theory of Groups</i>. New York: Macmillan.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Groups&rft.place=New+York&rft.pub=Macmillan&rft.date=1959&rft.aulast=Hall&rft.aufirst=Marshall&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrattini+subgroup" class="Z3988"></span> (See Chapter 10, especially Section 10.4.)</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol 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dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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