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The group of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436801.png" /> invertible matrices over an associative ring (cf. [[Associative rings and algebras|Associative rings and algebras]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436802.png" /> with a unit; the usual symbols are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436803.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436804.png" />. The general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436805.png" /> can also be defined as the automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436806.png" /> of the free right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436807.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436808.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g0436809.png" /> generators.
+
The general linear group of degree $n$ is
 +
the group of all $(n\times n)$ invertible matrices over an associative ring (cf.
 +
[[Associative rings and algebras|Associative rings and algebras]]) $K$
 +
with a unit; the usual symbols are $\def\GL{\textrm{GL}} \GL_n(K)$ or $\GL(n,K)$. The general linear
 +
group $\GL(n,K)$ can also be defined as the automorphism group $\textrm{Aut}_K(V)$ of the
 +
free right $K$-module $V$ with $n$ generators.
  
In research on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368010.png" /> its normal structure is of considerable interest. The centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368011.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368012.png" /> consists of scalar matrices with entries from the centre of the ring (cf. [[Centre of a ring|Centre of a ring]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368013.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368014.png" /> is commutative one defines the [[Special linear group|special linear group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368015.png" />, which consists of matrices with determinant 1. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368016.png" /> is a field, the [[Commutator subgroup|commutator subgroup]] of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368017.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368018.png" /> (apart from the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368020.png" />), and any [[Normal subgroup|normal subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368021.png" /> is either contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368022.png" /> or contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368023.png" />. In particular, the projective special linear group
+
In research on the group $\GL(n,K)$ its normal structure is of considerable
 +
interest. The centre $Z_n$ of the group $\GL(n,K)$ consists of scalar matrices
 +
with entries from the centre of the ring (cf.
 +
[[Centre of a ring|Centre of a ring]]) $K$. When $K$ is commutative
 +
one defines the
 +
[[Special linear group|special linear group]] $\def\SL{\textrm{SL}} \SL(n,K)$, which consists of
 +
matrices with determinant 1. When $K$ is a field, the
 +
[[Commutator subgroup|commutator subgroup]] of the group $\GL(n,K)$ coincides
 +
with $\SL(n,K)$ (apart from the case $n=2$, $|K| = 2$), and any
 +
[[Normal subgroup|normal subgroup]] of $\GL(n,K)$ is either contained in $Z_n$
 +
or contains $\SL(n,K)$. In particular, the projective special linear group
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368024.png" /></td> </tr></table>
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$$\def\PSL{\textrm{PSL}} \PSL(n,K) = \SL(n,K)/\SL(n,K)\cap Z_n$$
 +
is a simple group (apart from the cases $n=2$, $|K|=2,3$).
  
is a simple group (apart from the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368026.png" />).
+
If $K$ is a
 +
[[Skew-field|skew-field]] and $n>1$, any normal subgroup of $\GL(n,K)$ is
 +
either contained in $Z_n$ or contains the commutator subgroup $\SL^+(n,K)$ of $\GL(n,K)$
 +
generated by transvections (cf.
 +
[[Transvection|Transvection]]), and the quotient group $\SL^+(n,K)/\SL^+(n,K)\cap Z_n$ is
 +
simple. Also, there exists a natural isomorphism
 +
$$\GL(n,K)/\SL^+(n,K) \simeq K^*/[K^*,K^*],$$
 +
where $K^*$ is
 +
the multiplicative group of the skew-field $K$. If $K$ is
 +
finite-dimensional over its centre $k$, then the role of $\SL(n,K)$ is played
 +
by the group of all matrices from $\GL(n,K)$ with reduced norm 1. The groups
 +
$\SL(n,K)$ and $\SL^+(n,K)$ do not always coincide, although this is so if $k$ is a
 +
global field (see
 +
[[Kneser–Tits hypothesis|Kneser–Tits hypothesis]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368027.png" /> is a [[Skew-field|skew-field]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368028.png" />, any normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368029.png" /> is either contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368030.png" /> or contains the commutator subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368032.png" /> generated by transvections (cf. [[Transvection|Transvection]]), and the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368033.png" /> is simple. Also, there exists a natural isomorphism
+
The study of the normal structure of general linear groups over a ring
 +
$K$ is associated with
 +
[[Algebraic K-theory|algebraic $K$-theory]]. The group $\GL(n,K)$ over a
 +
general ring $K$ may contain numerous normal subgroups. For example,
 +
if $K$ is a commutative ring without zero divisors and with a finite
 +
number of generators, then $\GL(n,K)$ is a
 +
[[Residually-finite group|residually-finite group]], i.e. for each
 +
element $g$ there exists a normal subgroup $N_g$ of finite index not
 +
containing $g$. In the case $K=\Z$, the description of the normal
 +
subgroups of $\GL(n,\Z)$ is in fact equivalent to the
 +
[[Congruence problem|congruence problem]] for $\SL(n,\Z)$, since
 +
$$[\GL(n,\Z):\SL(n,\Z)] = 2,$$
 +
and any
 +
non-scalar normal subgroup of the group $\SL(n,\Z)$ for $n>2$ is a
 +
[[Congruence subgroup|congruence subgroup]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368034.png" /></td> </tr></table>
+
There is a deep analogy between the structure of general linear groups
 
+
and that of other classical groups. This analogy extends also to
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368035.png" /> is the multiplicative group of the skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368037.png" /> is finite-dimensional over its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368038.png" />, then the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368039.png" /> is played by the group of all matrices from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368040.png" /> with reduced norm 1. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368042.png" /> do not always coincide, although this is so if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368043.png" /> is a global field (see [[Kneser–Tits hypothesis|Kneser–Tits hypothesis]]).
+
simple algebraic groups and Lie groups.
 
 
The study of the normal structure of general linear groups over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368044.png" /> is associated with [[Algebraic K-theory|algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368045.png" />-theory]]. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368046.png" /> over a general ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368047.png" /> may contain numerous normal subgroups. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368048.png" /> is a commutative ring without zero divisors and with a finite number of generators, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368049.png" /> is a [[Residually-finite group|residually-finite group]], i.e. for each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368050.png" /> there exists a normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368051.png" /> of finite index not containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368052.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368053.png" />, the description of the normal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368054.png" /> is in fact equivalent to the [[Congruence problem|congruence problem]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368055.png" />, since
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368056.png" /></td> </tr></table>
 
 
 
and any non-scalar normal subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368057.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368058.png" /> is a [[Congruence subgroup|congruence subgroup]].
 
 
 
There is a deep analogy between the structure of general linear groups and that of other classical groups. This analogy extends also to simple algebraic groups and Lie groups.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"E. Artin,   "Geometric algebra" , Interscience (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Dieudonné,   "La géométrie des groups classiques" , Springer  (1955)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bass,   "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043680/g04368059.png" />-theory" , Benjamin  (1968)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Ar}}||valign="top"| E. Artin, "Geometric algebra", Interscience (1957) {{MR|0082463}}  {{ZBL|0077.02101}}
 +
|-
 +
|valign="top"|{{Ref|Ba}}||valign="top"| H. Bass, "Algebraic K-theory", Benjamin (1968) {{MR|0249491}}  {{ZBL|0174.30302}}
 +
|-
 +
|valign="top"|{{Ref|Di}}||valign="top"| J.A. Dieudonné, "La géométrie des groupes classiques", Springer (1955) {{MR|0072144}}  {{ZBL|0067.26104}}
 +
|-
 +
|}

Revision as of 22:32, 29 February 2012

2020 Mathematics Subject Classification: Primary: 20-XX Secondary: 15-XX [MSN][ZBL]

The general linear group of degree $n$ is the group of all $(n\times n)$ invertible matrices over an associative ring (cf. Associative rings and algebras) $K$ with a unit; the usual symbols are $\def\GL{\textrm{GL}} \GL_n(K)$ or $\GL(n,K)$. The general linear group $\GL(n,K)$ can also be defined as the automorphism group $\textrm{Aut}_K(V)$ of the free right $K$-module $V$ with $n$ generators.

In research on the group $\GL(n,K)$ its normal structure is of considerable interest. The centre $Z_n$ of the group $\GL(n,K)$ consists of scalar matrices with entries from the centre of the ring (cf. Centre of a ring) $K$. When $K$ is commutative one defines the special linear group $\def\SL{\textrm{SL}} \SL(n,K)$, which consists of matrices with determinant 1. When $K$ is a field, the commutator subgroup of the group $\GL(n,K)$ coincides with $\SL(n,K)$ (apart from the case $n=2$, $|K| = 2$), and any normal subgroup of $\GL(n,K)$ is either contained in $Z_n$ or contains $\SL(n,K)$. In particular, the projective special linear group

$$\def\PSL{\textrm{PSL}} \PSL(n,K) = \SL(n,K)/\SL(n,K)\cap Z_n$$ is a simple group (apart from the cases $n=2$, $|K|=2,3$).

If $K$ is a skew-field and $n>1$, any normal subgroup of $\GL(n,K)$ is either contained in $Z_n$ or contains the commutator subgroup $\SL^+(n,K)$ of $\GL(n,K)$ generated by transvections (cf. Transvection), and the quotient group $\SL^+(n,K)/\SL^+(n,K)\cap Z_n$ is simple. Also, there exists a natural isomorphism $$\GL(n,K)/\SL^+(n,K) \simeq K^*/[K^*,K^*],$$ where $K^*$ is the multiplicative group of the skew-field $K$. If $K$ is finite-dimensional over its centre $k$, then the role of $\SL(n,K)$ is played by the group of all matrices from $\GL(n,K)$ with reduced norm 1. The groups $\SL(n,K)$ and $\SL^+(n,K)$ do not always coincide, although this is so if $k$ is a global field (see Kneser–Tits hypothesis).

The study of the normal structure of general linear groups over a ring $K$ is associated with algebraic $K$-theory. The group $\GL(n,K)$ over a general ring $K$ may contain numerous normal subgroups. For example, if $K$ is a commutative ring without zero divisors and with a finite number of generators, then $\GL(n,K)$ is a residually-finite group, i.e. for each element $g$ there exists a normal subgroup $N_g$ of finite index not containing $g$. In the case $K=\Z$, the description of the normal subgroups of $\GL(n,\Z)$ is in fact equivalent to the congruence problem for $\SL(n,\Z)$, since $$[\GL(n,\Z):\SL(n,\Z)] = 2,$$ and any non-scalar normal subgroup of the group $\SL(n,\Z)$ for $n>2$ is a congruence subgroup.

There is a deep analogy between the structure of general linear groups and that of other classical groups. This analogy extends also to simple algebraic groups and Lie groups.

References

[Ar] E. Artin, "Geometric algebra", Interscience (1957) MR0082463 Zbl 0077.02101
[Ba] H. Bass, "Algebraic K-theory", Benjamin (1968) MR0249491 Zbl 0174.30302
[Di] J.A. Dieudonné, "La géométrie des groupes classiques", Springer (1955) MR0072144 Zbl 0067.26104
How to Cite This Entry:
General linear group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=General_linear_group&oldid=20768
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article