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<span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Characteristic_lattices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Characteristic_lattices"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Characteristic lattices</span> </div> </a> <ul id="toc-Characteristic_lattices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Characterizing_groups_by_their_subgroup_lattices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Characterizing_groups_by_their_subgroup_lattices"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Characterizing groups by their subgroup lattices</span> </div> </a> <ul id="toc-Characterizing_groups_by_their_subgroup_lattices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" 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decoding="async" width="400" height="333" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Dih4_subgroups_%28cycle_graphs%29.svg/600px-Dih4_subgroups_%28cycle_graphs%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Dih4_subgroups_%28cycle_graphs%29.svg/800px-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 lattice of subgroups 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>, with the subgroups represented by their <a href="/wiki/Cycle_graph_(algebra)" title="Cycle graph (algebra)">cycle graphs</a></figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>lattice of subgroups</b> of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is the <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> whose elements are the <a href="/wiki/Subgroup" title="Subgroup">subgroups</a> of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, with the <a href="/wiki/Partial_ordering" class="mw-redirect" title="Partial ordering">partial ordering</a> being <a href="/wiki/Set_inclusion" class="mw-redirect" title="Set inclusion">set inclusion</a>. In this lattice, the <a href="/wiki/Join_and_meet" title="Join and meet">join</a> of two subgroups is the subgroup <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generated</a> by their <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a>, and the <a href="/wiki/Join_and_meet" title="Join and meet">meet</a> of two subgroups is their <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_of_subgroups&action=edit&section=1" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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> has ten subgroups, counting itself and the <a href="/wiki/Trivial_group" title="Trivial group">trivial</a> subgroup. Five of the eight group elements generate subgroups of <a href="/wiki/Order_of_a_group" class="mw-redirect" title="Order of a group">order</a> two, and the other two non-<a href="/wiki/Identity_element" title="Identity element">identity</a> elements both generate the same <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a> subgroup of order four. In addition, there are two subgroups of the form <a href="/wiki/Klein_four-group" title="Klein four-group"><b>Z</b><sub>2</sub> × <b>Z</b><sub>2</sub></a>, generated by pairs of <span class="nowrap"><a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a>-two</span> elements. The lattice formed by these ten subgroups is shown in the illustration. </p><p>This example also shows that the lattice of all subgroups of a group is not a <a href="/wiki/Modular_lattice" title="Modular lattice">modular lattice</a> in general. Indeed, this particular lattice contains the forbidden "pentagon" N<sub>5</sub> as a <a href="/wiki/Sublattice" class="mw-redirect" title="Sublattice">sublattice</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_of_subgroups&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any <i>A</i>, <i>B</i>, and <i>C</i> subgroups of a group with <i>A</i> ≤ <i>C</i> (<i>A</i> a subgroup of <i>C</i>) then <i>AB</i> ∩ <i>C</i> = <i>A</i>(<i>B</i> ∩ <i>C</i>); the multiplication here is the <a href="/wiki/Product_of_subgroups" class="mw-redirect" title="Product of subgroups">product of subgroups</a>. This property has been called the <i>modular property of groups</i> (<a href="#CITEREFAschbacher2000">Aschbacher 2000</a>) or <i>(<a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind</a>'s) modular law</i> (<a href="#CITEREFRobinson1996">Robinson 1996</a>, <a href="#CITEREFCohn2000">Cohn 2000</a>). Since for two <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroups</a> the product is actually the <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">smallest subgroup containing</a> the two, the normal subgroups form a <a href="/wiki/Modular_lattice" title="Modular lattice">modular lattice</a>. </p><p>The <a href="/wiki/Lattice_theorem" class="mw-redirect" title="Lattice theorem">lattice theorem</a> establishes a <a href="/wiki/Galois_connection" title="Galois connection">Galois connection</a> between the lattice of subgroups of a group and that of its <a href="/wiki/Quotient_group" title="Quotient group">quotients</a>. </p><p>The <a href="/wiki/Zassenhaus_lemma" title="Zassenhaus lemma">Zassenhaus lemma</a> gives an <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphism</a> between certain combinations of quotients and products in the lattice of subgroups. </p><p>As groups are algebraic structures, it follows by a general Theorem (<a href="#CITEREFBurrisSankappanavar2011">Burris & Sankappanavar 2011</a>, p. 33) that their lattices of subgroups are algebraic lattices. This means that they are complete and compactly generated. However in general, there is no restriction on the possible sublattices of the lattice of subgroups, in the sense that every lattice is <a href="/wiki/Lattice_isomorphism" class="mw-redirect" title="Lattice isomorphism">isomorphic</a> to a sublattice of the subgroup lattice of some group. Furthermore, every <a href="/wiki/Finite_set" title="Finite set">finite</a> lattice is isomorphic to a sublattice of the subgroup lattice of some <a href="/wiki/Finite_group" title="Finite group">finite group</a> (<a href="#CITEREFSchmidt1994">Schmidt 1994</a>, p. 9). Every finite <a href="/wiki/Distributive_lattice" title="Distributive lattice">distributive lattice</a> is also isomorphic to the normal subgroup lattice of some group (<a href="#CITEREFSilcock1977">Silcock 1977</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Characteristic_lattices">Characteristic lattices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_of_subgroups&action=edit&section=3" title="Edit section: Characteristic lattices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Subgroups with certain properties form lattices, but other properties do not. </p> <ul><li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroups</a> always form a <a href="/wiki/Modular_lattice" title="Modular lattice">modular lattice</a>. In fact, the essential property that guarantees that the lattice is modular is that subgroups commute with each other, i.e. that they are <a href="/wiki/Quasinormal_subgroup" title="Quasinormal subgroup">quasinormal subgroups</a>.</li> <li><a href="/wiki/Nilpotent_group" title="Nilpotent group">Nilpotent</a> normal subgroups form a lattice, which is (part of) the content of <a href="/wiki/Fitting%27s_theorem" title="Fitting's theorem">Fitting's theorem</a>.</li> <li>A class of groups is called a <i>Fitting class</i> if it is closed under isomorphism, <a href="/wiki/Subnormal_subgroup" title="Subnormal subgroup">subnormal subgroups</a>, and products of subnormal subgroups. For any Fitting class <i>F</i>, both the subnormal <i>F</i>-subgroups and the normal <i>F</i>-subgroups form lattices. This generalizes the above with <i>F</i> the class of nilpotent groups, and another example is with <i>F</i> the class of <a href="/wiki/Solvable_group" title="Solvable group">solvable groups</a>.</li> <li><a href="/wiki/Central_subgroup" title="Central subgroup">Central subgroups</a> form a lattice.</li></ul> <p>However, neither finite subgroups nor <a href="/wiki/Torsion_group" title="Torsion group">torsion</a> subgroups form a lattice: for instance, the <a href="/wiki/Free_product" title="Free product">free product</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 \mathbf {Z} /2\mathbf {Z} *\mathbf {Z} /2\mathbf {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} /2\mathbf {Z} *\mathbf {Z} /2\mathbf {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/084fbe876307f176b7f7d899a6828243bd6633db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.38ex; height:2.843ex;" alt="{\displaystyle \mathbf {Z} /2\mathbf {Z} *\mathbf {Z} /2\mathbf {Z} }"></span> is generated by two <a href="/wiki/Torsion_element" class="mw-redirect" title="Torsion element">torsion elements</a>, but is <a href="/wiki/Infinite_group" title="Infinite group">infinite</a> and contains elements of infinite order. </p><p>The fact that normal subgroups form a modular lattice is a particular case of a more general result, namely that in any <a href="/wiki/Maltsev_variety" class="mw-redirect" title="Maltsev variety">Maltsev variety</a> (of which groups are an example), the <a href="/wiki/Congruence_lattice" class="mw-redirect" title="Congruence lattice">lattice of congruences</a> is modular (<a href="#CITEREFKearnesKiss2013">Kearnes & Kiss 2013</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Characterizing_groups_by_their_subgroup_lattices">Characterizing groups by their subgroup lattices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_of_subgroups&action=edit&section=4" title="Edit section: Characterizing groups by their subgroup lattices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Lattice_theory" class="mw-redirect" title="Lattice theory">Lattice-theoretic</a> information about the lattice of subgroups can sometimes be used to infer information about the original group, an idea that goes back to the work of <a href="/wiki/%C3%98ystein_Ore" title="Øystein Ore">Øystein Ore</a> (<a href="#CITEREFOre1937">1937</a>, <a href="#CITEREFOre1938">1938</a>). For instance, as Ore <a href="/wiki/Mathematical_proof" title="Mathematical proof">proved</a>, a group is <a href="/wiki/Locally_cyclic_group" title="Locally cyclic group">locally cyclic</a> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> its lattice of subgroups is <a href="/wiki/Distributive_lattice" title="Distributive lattice">distributive</a>. If additionally the lattice satisfies the <a href="/wiki/Ascending_chain_condition" title="Ascending chain condition">ascending chain condition</a>, then the group is cyclic. </p><p>Groups whose lattice of subgroups is a <a href="/wiki/Complemented_lattice" title="Complemented lattice">complemented lattice</a> are called <a href="/wiki/Complemented_group" title="Complemented group">complemented groups</a> (<a href="#CITEREFZacher1953">Zacher 1953</a>), and groups whose lattice of subgroups are <a href="/wiki/Modular_lattice" title="Modular lattice">modular lattices</a> are called <a href="/wiki/Iwasawa_group" title="Iwasawa group">Iwasawa groups</a> or modular groups (<a href="#CITEREFIwasawa1941">Iwasawa 1941</a>). Lattice-theoretic characterizations of this type also exist for <a href="/wiki/Solvable_group" title="Solvable group">solvable groups</a> and <a href="/wiki/Perfect_group" title="Perfect group">perfect groups</a> (<a href="#CITEREFSuzuki1951">Suzuki 1951</a>). </p> <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=Lattice_of_subgroups&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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="CITEREFAschbacher2000" class="citation book cs1">Aschbacher, M. (2000). <i>Finite Group Theory</i>. Cambridge University Press. p. 6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-78675-1" title="Special:BookSources/978-0-521-78675-1"><bdi>978-0-521-78675-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+Group+Theory&rft.pages=6&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0-521-78675-1&rft.aulast=Aschbacher&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaer1939" class="citation journal cs1">Baer, Reinhold (1939). "The significance of the system of subgroups for the structure of the group". <i><a href="/wiki/American_Journal_of_Mathematics" title="American Journal of Mathematics">American Journal of Mathematics</a></i>. <b>61</b> (1). The Johns Hopkins University Press: 1–44. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2371383">10.2307/2371383</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2371383">2371383</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Mathematics&rft.atitle=The+significance+of+the+system+of+subgroups+for+the+structure+of+the+group&rft.volume=61&rft.issue=1&rft.pages=1-44&rft.date=1939&rft_id=info%3Adoi%2F10.2307%2F2371383&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2371383%23id-name%3DJSTOR&rft.aulast=Baer&rft.aufirst=Reinhold&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohn2000" class="citation book cs1">Cohn, Paul Moritz (2000). <i>Classic algebra</i>. Wiley. p. 248. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-87731-8" title="Special:BookSources/978-0-471-87731-8"><bdi>978-0-471-87731-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classic+algebra&rft.pages=248&rft.pub=Wiley&rft.date=2000&rft.isbn=978-0-471-87731-8&rft.aulast=Cohn&rft.aufirst=Paul+Moritz&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIwasawa1941" class="citation cs2">Iwasawa, Kenkiti (1941), "Über die endlichen Gruppen und die Verbände ihrer Untergruppen", <i>J. Fac. Sci. Imp. Univ. Tokyo. Sect. I.</i>, <b>4</b>: 171–199, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0005721">0005721</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Fac.+Sci.+Imp.+Univ.+Tokyo.+Sect.+I.&rft.atitle=%C3%9Cber+die+endlichen+Gruppen+und+die+Verb%C3%A4nde+ihrer+Untergruppen&rft.volume=4&rft.pages=171-199&rft.date=1941&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0005721%23id-name%3DMR&rft.aulast=Iwasawa&rft.aufirst=Kenkiti&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKearnesKiss2013" class="citation book cs1">Kearnes, Keith; Kiss, Emil W. (2013). <i>The Shape of Congruence Lattices</i>. American Mathematical Soc. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-8323-5" title="Special:BookSources/978-0-8218-8323-5"><bdi>978-0-8218-8323-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Shape+of+Congruence+Lattices&rft.pages=3&rft.pub=American+Mathematical+Soc.&rft.date=2013&rft.isbn=978-0-8218-8323-5&rft.aulast=Kearnes&rft.aufirst=Keith&rft.au=Kiss%2C+Emil+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOre1937" class="citation journal cs1"><a href="/wiki/%C3%98ystein_Ore" title="Øystein Ore">Ore, Øystein</a> (1937). "Structures and group theory. I". <i>Duke Mathematical Journal</i>. <b>3</b> (2): 149–174. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2FS0012-7094-37-00311-9">10.1215/S0012-7094-37-00311-9</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1545977">1545977</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Duke+Mathematical+Journal&rft.atitle=Structures+and+group+theory.+I&rft.volume=3&rft.issue=2&rft.pages=149-174&rft.date=1937&rft_id=info%3Adoi%2F10.1215%2FS0012-7094-37-00311-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1545977%23id-name%3DMR&rft.aulast=Ore&rft.aufirst=%C3%98ystein&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOre1938" class="citation journal cs1"><a href="/wiki/%C3%98ystein_Ore" title="Øystein Ore">Ore, Øystein</a> (1938). "Structures and group theory. II". <i>Duke Mathematical Journal</i>. <b>4</b> (2): 247–269. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2FS0012-7094-38-00419-3">10.1215/S0012-7094-38-00419-3</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F100155">10338.dmlcz/100155</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1546048">1546048</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Duke+Mathematical+Journal&rft.atitle=Structures+and+group+theory.+II&rft.volume=4&rft.issue=2&rft.pages=247-269&rft.date=1938&rft_id=info%3Ahdl%2F10338.dmlcz%2F100155&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1546048%23id-name%3DMR&rft_id=info%3Adoi%2F10.1215%2FS0012-7094-38-00419-3&rft.aulast=Ore&rft.aufirst=%C3%98ystein&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobinson1996" class="citation book cs1">Robinson, Derek (1996). <i>A Course in the Theory of Groups</i>. Springer Science & Business Media. p. 15. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94461-6" title="Special:BookSources/978-0-387-94461-6"><bdi>978-0-387-94461-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+the+Theory+of+Groups&rft.pages=15&rft.pub=Springer+Science+%26+Business+Media&rft.date=1996&rft.isbn=978-0-387-94461-6&rft.aulast=Robinson&rft.aufirst=Derek&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRottlaender1928" class="citation journal cs1">Rottlaender, Ada (1928). "Nachweis der Existenz nicht-isomorpher Gruppen von gleicher Situation der Untergruppen". <i><a href="/wiki/Mathematische_Zeitschrift" title="Mathematische Zeitschrift">Mathematische Zeitschrift</a></i>. <b>28</b> (1): 641–653. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01181188">10.1007/BF01181188</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120596994">120596994</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Zeitschrift&rft.atitle=Nachweis+der+Existenz+nicht-isomorpher+Gruppen+von+gleicher+Situation+der+Untergruppen&rft.volume=28&rft.issue=1&rft.pages=641-653&rft.date=1928&rft_id=info%3Adoi%2F10.1007%2FBF01181188&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120596994%23id-name%3DS2CID&rft.aulast=Rottlaender&rft.aufirst=Ada&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt1994" class="citation book cs1">Schmidt, Roland (1994). <i>Subgroup Lattices of Groups</i>. Expositions in Math. Vol. 14. Walter de Gruyter. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-011213-9" title="Special:BookSources/978-3-11-011213-9"><bdi>978-3-11-011213-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Subgroup+Lattices+of+Groups&rft.series=Expositions+in+Math&rft.pub=Walter+de+Gruyter&rft.date=1994&rft.isbn=978-3-11-011213-9&rft.aulast=Schmidt&rft.aufirst=Roland&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://www.ams.org/bull/1996-33-04/S0273-0979-96-00676-3/S0273-0979-96-00676-3.pdf">Review</a> by Ralph Freese in Bull. AMS <b>33</b> (4): 487–492.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSuzuki1951" class="citation journal cs1"><a href="/wiki/Michio_Suzuki_(mathematician)" title="Michio Suzuki (mathematician)">Suzuki, Michio</a> (1951). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990375">"On the lattice of subgroups of finite groups"</a>. <i><a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a></i>. <b>70</b> (2). American Mathematical Society: 345–371. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990375">10.2307/1990375</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990375">1990375</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=On+the+lattice+of+subgroups+of+finite+groups&rft.volume=70&rft.issue=2&rft.pages=345-371&rft.date=1951&rft_id=info%3Adoi%2F10.2307%2F1990375&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1990375%23id-name%3DJSTOR&rft.aulast=Suzuki&rft.aufirst=Michio&rft_id=https%3A%2F%2Fdoi.org%2F10.2307%252F1990375&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSuzuki1956" class="citation book cs1"><a href="/wiki/Michio_Suzuki_(mathematician)" title="Michio Suzuki (mathematician)">Suzuki, Michio</a> (1956). <i>Structure of a Group and the Structure of its Lattice of Subgroups</i>. Berlin: Springer Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Structure+of+a+Group+and+the+Structure+of+its+Lattice+of+Subgroups&rft.place=Berlin&rft.pub=Springer+Verlag&rft.date=1956&rft.aulast=Suzuki&rft.aufirst=Michio&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYakovlev1974" class="citation journal cs1">Yakovlev, B. V. (1974). "Conditions under which a lattice is isomorphic to a lattice of subgroups of a group". <i>Algebra and Logic</i>. <b>13</b> (6): 400–412. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01462952">10.1007/BF01462952</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119943975">119943975</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Algebra+and+Logic&rft.atitle=Conditions+under+which+a+lattice+is+isomorphic+to+a+lattice+of+subgroups+of+a+group&rft.volume=13&rft.issue=6&rft.pages=400-412&rft.date=1974&rft_id=info%3Adoi%2F10.1007%2FBF01462952&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119943975%23id-name%3DS2CID&rft.aulast=Yakovlev&rft.aufirst=B.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilcock1977" class="citation journal cs1">Silcock, Howard L. (1977). <a rel="nofollow" class="external text" href="https://link.springer.com/content/pdf/10.1007/BF02485445.pdf">"Generalized wreath products and the lattice of normal subgroups of a group"</a> <span class="cs1-format">(PDF)</span>. <i>Algebra Universalis</i>. <b>7</b>: 361–372.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Algebra+Universalis&rft.atitle=Generalized+wreath+products+and+the+lattice+of+normal+subgroups+of+a+group&rft.volume=7&rft.pages=361-372&rft.date=1977&rft.aulast=Silcock&rft.aufirst=Howard+L.&rft_id=https%3A%2F%2Flink.springer.com%2Fcontent%2Fpdf%2F10.1007%2FBF02485445.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZacher1953" class="citation journal cs1">Zacher, Giovanni (1953). <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=RSMUP_1953__22__113_0">"Caratterizzazione dei gruppi risolubili d'ordine finito complementati"</a>. <i><a href="/wiki/Rendiconti_del_Seminario_Matematico_della_Universit%C3%A0_di_Padova" title="Rendiconti del Seminario Matematico della Università di Padova">Rendiconti del Seminario Matematico della Università di Padova</a></i>. <b>22</b>: 113–122. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0041-8994">0041-8994</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0057878">0057878</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rendiconti+del+Seminario+Matematico+della+Universit%C3%A0+di+Padova&rft.atitle=Caratterizzazione+dei+gruppi+risolubili+d%27ordine+finito+complementati&rft.volume=22&rft.pages=113-122&rft.date=1953&rft.issn=0041-8994&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0057878%23id-name%3DMR&rft.aulast=Zacher&rft.aufirst=Giovanni&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DRSMUP_1953__22__113_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurrisSankappanavar2011" class="citation book cs1">Burris, S.; Sankappanavar, H. P. (2011). <i>A Course in Universal Algebra</i>. Graduate Texts in Mathematics. Vol. 78. Springer Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4613-8132-7" title="Special:BookSources/978-1-4613-8132-7"><bdi>978-1-4613-8132-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Universal+Algebra&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer+Verlag&rft.date=2011&rft.isbn=978-1-4613-8132-7&rft.aulast=Burris&rft.aufirst=S.&rft.au=Sankappanavar%2C+H.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+of+subgroups" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_of_subgroups&action=edit&section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://planetmath.org/latticeofsubgroups">PlanetMath entry on lattice of subgroups</a></li> <li>Example: <a href="https://en.wikiversity.org/wiki/Symmetric_group_S4#Lattice_of_subgroups" class="extiw" title="v:Symmetric group S4">Lattice of subgroups of the symmetric group S4</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Lattice_of_subgroups&oldid=1245756890">https://en.wikipedia.org/w/index.php?title=Lattice_of_subgroups&oldid=1245756890</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Lattice_theory" title="Category:Lattice theory">Lattice theory</a></li><li><a href="/wiki/Category:Group_theory" title="Category:Group theory">Group theory</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 14 September 2024, at 22:13<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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