General linear group

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