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<title>Linear group</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Linear group</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>matrix group</b> is a <a href="Group_(mathematics)" title="Group (mathematics)">group</a> <i>G</i> consisting of <a href="Invertible_matrix" title="Invertible matrix">invertible</a> <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> over a specified <a href="Field_(mathematics)" title="Field (mathematics)">field</a> <i>K</i>, with the operation of <a href="Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>. A <b>linear group</b> is a group that is <a href="Group_isomorphism" title="Group isomorphism">isomorphic</a> to a matrix group (that is, admitting a <a href="Faithful_representation" title="Faithful representation">faithful</a>, finite-dimensional <a href="Group_representation" title="Group representation">representation</a> over <i>K</i>).

</p><p>Any <a href="Finite_group" title="Finite group">finite group</a> is linear, because it can be realized by <a href="Permutation_matrices" class="mw-redirect" title="Permutation matrices">permutation matrices</a> using <a href="Cayley's_theorem" title="Cayley's theorem">Cayley's theorem</a>. Among <a href="Infinite_group_theory" class="mw-redirect" title="Infinite group theory">infinite groups</a>, linear groups form an interesting and tractable class. Examples of groups that are not linear include groups which are "too big" (for example, the group of permutations of an infinite set), or which exhibit some pathological behavior (for example, <a href="Finitely_generated_group" title="Finitely generated group">finitely generated</a> infinite <a href="Torsion_group" title="Torsion group">torsion groups</a>).

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition_and_basic_examples">Definition and basic examples</h2></div>

<p>A group <i>G</i> is said to be <i>linear</i> if there exists a field <i>K</i>, an <a href="Integer" title="Integer">integer</a> <i>d</i> and an <a href="Injectivity" class="mw-redirect" title="Injectivity">injective</a> <a href="Group_homomorphism" title="Group homomorphism">homomorphism</a> from <i>G</i> to the <a href="General_linear_group" title="General linear group">general linear group</a> GL<sub><i>d</i> </sub>(<i>K</i>) (a faithful linear <a href="Group_representation" title="Group representation">representation</a> of dimension <i>d</i> over <i>K</i>): if needed one can mention the field and dimension by saying that <i>G</i> is <i>linear of degree d over K</i>. Basic instances are groups which are defined as <a href="Subgroup" title="Subgroup">subgroups</a> of a linear group, for example:

</p>

<ol><li>The group GL<sub><i>n</i></sub>(<i>K</i>) itself;</li>

<li>The <a href="Special_linear_group" title="Special linear group">special linear group</a> SL<sub><i>n</i></sub>(<i>K</i>) (the subgroup of matrices with <a href="Determinant" title="Determinant">determinant</a> 1);</li>

<li>The group of invertible upper (or lower) <a href="Triangular_matrix" title="Triangular matrix">triangular matrices</a></li>

<li>If <i>g<sub>i</sub></i> is a collection of elements in GL<sub><i>n</i></sub>(<i>K</i>) <a href="Index_set" title="Index set">indexed</a> by a set <i>I</i>, then the subgroup generated by the <i>g<sub>i</sub></i> is a linear group.</li></ol>

<p>In the study of <a href="Lie_group" title="Lie group">Lie groups</a>, it is sometimes pedagogically convenient to restrict attention to Lie groups that can be faithfully represented over the field of <a href="Complex_number" title="Complex number">complex numbers</a>. (Some authors require that the group be represented as a <i>closed</i> subgroup of the GL<sub><i>n</i></sub>(<b>C</b>).) Books that follow this approach include Hall (2015)<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> and Rossmann (2002).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Classes_of_linear_groups">Classes of linear groups</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Classical_groups_and_related_examples">Classical groups and related examples</h3></div>

<p>The so-called <a href="Classical_group" title="Classical group">classical groups</a> generalize the examples 1 and 2 above. They arise as <a href="Linear_algebraic_group" title="Linear algebraic group">linear algebraic groups</a>, that is, as subgroups of GL<sub><i>n</i></sub> defined by a finite number of equations. Basic examples are <a href="Orthogonal_group" title="Orthogonal group">orthogonal</a>, <a href="Unitary_group" title="Unitary group">unitary</a> and <a href="Symplectic_group" title="Symplectic group">symplectic</a> groups but it is possible to construct more using <a href="Division_algebra" title="Division algebra">division algebras</a> (for example the <a href="Unit_group" class="mw-redirect" title="Unit group">unit group</a> of a <a href="Quaternion_algebra" title="Quaternion algebra">quaternion algebra</a> is a classical group). Note that the <a href="Projective_group" class="mw-redirect" title="Projective group">projective groups</a> associated to these groups are also linear, though less obviously. For example, the group PSL<sub>2</sub>(<b>R</b>) is not a group of 2 × 2 matrices, but it has a faithful representation as 3 × 3 matrices (the <a href="Adjoint_representation" title="Adjoint representation">adjoint representation</a>), which can be used in the general case.

</p><p>Many <a href="Lie_group" title="Lie group">Lie groups</a> are linear, but not all of them. The <a href="SL2(R)#Topology_and_universal_cover" title="SL2(R)">universal cover of SL<sub>2</sub>(<b>R</b>)</a> is not linear, as are many <a href="Solvable_group" title="Solvable group">solvable groups</a>, for instance the <a href="Quotient_group" title="Quotient group">quotient</a> of the <a href="Heisenberg_group" title="Heisenberg group">Heisenberg group</a> by a <a href="Central_subgroup" title="Central subgroup">central</a> cyclic subgroup.

</p><p><a href="Discrete_subgroup" class="mw-redirect" title="Discrete subgroup">Discrete subgroups</a> of classical Lie groups (for example <a href="Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">lattices</a> or <a href="Thin_group_(algebraic_group_theory)" title="Thin group (algebraic group theory)">thin groups</a>) are also examples of interesting linear groups.

</p>

<div class="mw-heading mw-heading3"><h3 id="Finite_groups">Finite groups</h3></div>

<p>A finite group <i>G</i> of <a href="Order_(group_theory)" title="Order (group theory)">order</a> <i>n</i> is linear of degree at most <i>n</i> over any field <i>K</i>. This statement is sometimes called Cayley's theorem, and simply results from the fact that the action of <i>G</i> on the <a href="Group_ring" title="Group ring">group ring</a> <i>K</i>[<i>G</i>] by left (or right) multiplication is linear and faithful. The <a href="Group_of_Lie_type" title="Group of Lie type">finite groups of Lie type</a> (classical groups over finite fields) are an important family of finite <a href="Simple_group" title="Simple group">simple groups</a>, as they take up most of the slots in the <a href="Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Finitely_generated_matrix_groups">Finitely generated matrix groups</h3></div>

<p>While example 4 above is too general to define a distinctive class (it includes all linear groups), restricting to a finite index set <i>I</i>, that is, to <a href="Finitely_generated_group" title="Finitely generated group">finitely generated groups</a> allows to construct many interesting examples. For example:

</p>

<ul><li>The <a href="Ping-pong_lemma" title="Ping-pong lemma">ping-pong lemma</a> can be used to construct many examples of linear groups which are <a href="Free_group" title="Free group">free groups</a> (for instance the group generated by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}{}_{2}^{1}\,_{1}^{0}{\bigr )},\,{\bigl (}{}_{0}^{1}\,_{1}^{2}{\bigr )}}">

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<annotation encoding="application/x-tex">{\displaystyle {\bigl (}{}_{2}^{1}\,_{1}^{0}{\bigr )},\,{\bigl (}{}_{0}^{1}\,_{1}^{2}{\bigr )}}</annotation>

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</math></span><img src="./3b2f4d0d6f2eb85b43ce72588bde2d68a831345a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.672ex; height:3.176ex;" alt="{\displaystyle {\bigl (}{}_{2}^{1}\,_{1}^{0}{\bigr )},\,{\bigl (}{}_{0}^{1}\,_{1}^{2}{\bigr )}}" loading="lazy"></span> is free).</li>

<li><a href="Arithmetic_group" title="Arithmetic group">Arithmetic groups</a> are known to be finitely generated. On the other hand, it is a difficult problem to find an explicit set of generators for a given arithmetic group.</li>

<li><a href="Braid_group" title="Braid group">Braid groups</a> (which are defined as a <a href="Finitely_presented_group" class="mw-redirect" title="Finitely presented group">finitely presented group</a>) have faithful linear representation on a <a href="Dimension_(vector_space)" title="Dimension (vector space)">finite-dimensional</a> complex vector space where the generators act by explicit matrices.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The <a href="Mapping_class_group_of_a_surface" title="Mapping class group of a surface">mapping class group</a> of a genus 2 surface is also known to be linear.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Examples_from_geometry">Examples from geometry</h3></div>

<p>In some cases the <a href="Fundamental_group" title="Fundamental group">fundamental group</a> of a <a href="Manifold" title="Manifold">manifold</a> can be shown to be linear by using representations coming from a geometric structure. For example, all <a href="Closed_surface" class="mw-redirect" title="Closed surface">closed surfaces</a> of <a href="Genus_(mathematics)" title="Genus (mathematics)">genus</a> at least 2 are hyperbolic <a href="Riemann_surface" title="Riemann surface">Riemann surfaces</a>. Via the <a href="Uniformization_theorem" title="Uniformization theorem">uniformization theorem</a> this gives rise to a representation of its fundamental group in the <a href="Isometry_group" title="Isometry group">isometry group</a> of the <a href="Hyperbolic_plane" class="mw-redirect" title="Hyperbolic plane">hyperbolic plane</a>, which is isomorphic to PSL<sub>2</sub>(<b>R</b>) and this realizes the fundamental group as a <a href="Fuchsian_group" title="Fuchsian group">Fuchsian group</a>. A generalization of this construction is given by the notion of a <a href="(G%2CX)-structure" class="mw-redirect" title="(G,X)-structure">(<i>G</i>,<i>X</i>)-structure</a> on a manifold.

</p><p>Another example is the fundamental group of <a href="Seifert_manifold" class="mw-redirect" title="Seifert manifold">Seifert manifolds</a>. On the other hand, it is not known whether all fundamental groups of 3–manifolds are linear.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<p>While linear groups are a vast class of examples, among all infinite groups they are distinguished by many remarkable properties. Finitely generated linear groups have the following properties:

</p>

<ul><li>They are <a href="Residually_finite_group" title="Residually finite group">residually finite</a>;</li>

<li><a href="Burnside_problem" title="Burnside problem">Burnside's theorem</a>: a <a href="Torsion_group" title="Torsion group">torsion</a> group of finite <a href="Torsion_group" title="Torsion group">exponent</a> which is linear over a field of characteristic 0 must be finite;<sup id="cite_ref-FOOTNOTEWehrfritz197315_6-0" class="reference"><a href="#cite_note-FOOTNOTEWehrfritz197315-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li>

<li>Schur's theorem: a <a href="Torsion_group" title="Torsion group">torsion</a> linear group is <a href="Locally_finite_group" title="Locally finite group">locally finite</a>. In particular, if it is finitely generated then it is finite.<sup id="cite_ref-FOOTNOTEWehrfritz197357_7-0" class="reference"><a href="#cite_note-FOOTNOTEWehrfritz197357-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li>

<li>Selberg's lemma: any finitely generated linear group contains a <a href="Torsion-free_group" class="mw-redirect" title="Torsion-free group">torsion-free</a> subgroup of finite <a href="Index_of_a_subgroup" title="Index of a subgroup">index</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul>

<p>The <a href="Tits_alternative" title="Tits alternative">Tits alternative</a> states that a linear group either contains a non-abelian free group or else is <a href="Virtually" title="Virtually">virtually</a> solvable (that is, contains a <a href="Solvable_group" title="Solvable group">solvable group</a> of finite index). This has many further consequences, for example:

</p>

<ul><li>the <a href="Dehn_function" title="Dehn function">Dehn function</a> of a finitely generated linear group can only be either polynomial or exponential;</li>

<li>an <a href="Amenable_group" title="Amenable group">amenable</a> linear group is virtually solvable, in particular <a href="Elementary_amenable" class="mw-redirect" title="Elementary amenable">elementary amenable</a>;</li>

<li>the <a href="Von_Neumann_conjecture" title="Von Neumann conjecture">von Neumann conjecture</a> is true for linear groups.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Examples_of_non-linear_groups">Examples of non-linear groups</h2></div>

<p>It is not hard to give infinitely generated examples of non-linear groups: for example the infinite abelian group (<b>Z</b>/2<b>Z</b>)<sup><b>N</b></sup> x (<b>Z</b>/3<b>Z</b>)<sup><b>N</b></sup> cannot be linear.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Since the <a href="Symmetric_group" title="Symmetric group">symmetric group</a> on an infinite set contains this group it is also not linear. Finding finitely generated examples is subtler and usually requires the use of one of the properties listed above.

</p>

<ul><li>Since any finitely linear group is residually finite, it cannot be both simple and infinite. Thus finitely generated infinite simple groups, for example <a href="Thompson_groups" title="Thompson groups">Thompson's group</a> <i>F</i>, and the quotient of <a href="Higman's_group" class="mw-redirect" title="Higman's group">Higman's group</a> by a maximal proper normal subgroup, are not linear.</li>

<li>By the corollary to the Tits alternative mentioned above, groups of intermediate growth such as <a href="Grigorchuk's_group" class="mw-redirect" title="Grigorchuk's group">Grigorchuk's group</a> are not linear.</li>

<li>Again by the Tits alternative, as mentioned above all counterexamples to the <a href="Von_Neumann_conjecture" title="Von Neumann conjecture">von Neumann conjecture</a> are not linear. This includes <a href="Thompson_groups" title="Thompson groups">Thompson's group</a> <i>F</i> and <a href="Tarski_monster_group" title="Tarski monster group">Tarski monster groups</a>.</li>

<li>By Burnside's theorem, infinite, finitely generated torsion groups such as <a href="Tarski_monster_group" title="Tarski monster group">Tarski monster groups</a> cannot be linear.</li>

<li>There are examples of <a href="Hyperbolic_group" title="Hyperbolic group">hyperbolic groups</a> which are not linear, obtained as quotients of lattices in the Lie groups Sp(<i>n</i>, 1).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li>

<li>The <a href="Outer_automorphism_group" title="Outer automorphism group">outer automorphism group</a> <a href="Out(Fn)" title="Out(Fn)">Out(F<sub><i>n</i></sub>)</a> of the free group is known not to be linear for <i>n</i> at least 4.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li>

<li>In contrast with the case of braid groups, it is an <a href="Open_problem" title="Open problem">open question</a> whether the mapping class group of a surface of genus > 2 is linear.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Representation_theory">Representation theory</h2></div>

<p>Once a group has been established to be linear it is interesting to try to find "optimal" faithful linear representations for it, for example of the lowest possible dimension, or even to try to classify all its linear representations (including those which are not faithful). These questions are the object of <a href="Representation_theory" title="Representation theory">representation theory</a>. Salient parts of the theory include:

</p>

<ul><li><a href="Representation_theory_of_finite_groups" title="Representation theory of finite groups">Representation theory of finite groups</a>;</li>

<li><a href="Representation_theory_of_Lie_groups" class="mw-redirect" title="Representation theory of Lie groups">Representation theory of Lie groups</a> and more generally linear algebraic groups.</li></ul>

<p>The representation theory of infinite finitely generated groups is in general mysterious; the object of interest in this case are the <a href="Character_variety" title="Character variety">character varieties</a> of the group, which are well understood only in very few cases, for example free groups, surface groups and more generally lattices in Lie groups (for example through Margulis' <a href="Superrigidity" title="Superrigidity">superrigidity</a> theorem and other rigidity results).

</p>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFRossmann2002">Rossmann (2002)</a></span>

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</style><cite id="CITEREFStephen_J._Bigelow2000" class="citation cs2">Stephen J. Bigelow (December 13, 2000), <a rel="nofollow" class="external text" href="https://www.ams.org/jams/2001-14-02/S0894-0347-00-00361-1/S0894-0347-00-00361-1.pdf">"Braid groups are linear"</a> <span class="cs1-format">(PDF)</span>, <i>Journal of the American Mathematical Society</i>, <b>14</b> (2): <span class="nowrap">471–</span>486, <a href="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/10.1090%2FS0894-0347-00-00361-1">10.1090/S0894-0347-00-00361-1</a></span>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18936096">18936096</a></cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFBigelowBudney2001" class="citation cs2">Bigelow, Stephen J.; Budney, Ryan D. (2001), "The mapping class group of a genus two surface is linear", <i>Algebraic and Geometric Topology</i>, <b>1</b> (2): <span class="nowrap">699–</span>708, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0010310">math/0010310</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fagt.2001.1.699">10.2140/agt.2001.1.699</a></cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFAschenbrennerFriedlWilton2015" class="citation book cs1">Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry (2015). <a rel="nofollow" class="external text" href="http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/3-manifold-groups-final-version-031115"><i>3–manifolds groups</i></a>. EMS Series of Lectures in Mathematics. European Math. Soc. Section 9.6.</cite></span>

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<li id="cite_note-FOOTNOTEWehrfritz197315-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWehrfritz197315_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWehrfritz1973">Wehrfritz 1973</a>, p. 15.</span>

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<li id="cite_note-FOOTNOTEWehrfritz197357-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWehrfritz197357_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWehrfritz1973">Wehrfritz 1973</a>, p. 57.</span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFAlperin1987" class="citation journal cs1"><a href="Roger_C._Alperin" title="Roger C. Alperin">Alperin, Roger C.</a> (1987). "An Elementary Account Of Selberg's Lemma". <i>L'Enseignement Mathématique</i>. <b>33</b>.</cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">This follows from <a href="#CITEREFWehrfritz1973">Wehrfritz (1973</a>, Theorem 2.2).</span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFBestvina2004" class="citation web cs1">Bestvina, Mladen (2004). <a rel="nofollow" class="external text" href="http://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf">"Questions in Geometric Group Theory"</a> <span class="cs1-format">(PDF)</span>. Question 1.15<span class="reference-accessdate">. Retrieved <span class="nowrap">17 August</span> 2016</span>.</cite></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFFormanekProcesi1992" class="citation journal cs1">Formanek, E.; Procesi, C. (1992). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-8693%2892%2990029-l">"The automorphism group of a free group is not linear"</a>. <i>J. Algebra</i>. <b>149</b> (2): <span class="nowrap">494–</span>499. <a href="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/10.1016%2F0021-8693%2892%2990029-l">10.1016/0021-8693(92)90029-l</a></span>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFHall2015" class="citation cs2">Hall, Brian C. (2015), <i>Lie Groups, Lie Algebras, and Representations: An Elementary Introduction</i>, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3319134666</bdi></cite>.</li>

<li><cite id="CITEREFRossmann2002" class="citation cs2">Rossmann, Wulf (2002), <i>Lie Groups: An Introduction through Linear Groups</i>, Oxford Graduate Texts in Mathematics, Oxford University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780198596837</bdi></cite>.</li>

<li><cite id="CITEREFSuprnenko1976" class="citation book cs1">Suprnenko, D.A. (1976). <i>Matrix groups</i>. Translations of mathematical monographs. Vol. 45. <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-1595-4</bdi>.</cite></li>

<li><cite id="CITEREFWehrfritz1973" class="citation book cs1">Wehrfritz, B.A.F. (1973). <i>Infinite linear groups</i>. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 76. Springer-Verlag.</cite></li></ul></div><div><div class="zim-footer">

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