# General linear group

2010 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 subgroup 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.

How to Cite This Entry:
General linear group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=General_linear_group&oldid=35370
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article