General linear group

The group of all invertible matrices over an associative ring (cf. Associative rings and algebras) with a unit; the usual symbols are or . The general linear group can also be defined as the automorphism group of the free right -module with generators.

In research on the group its normal structure is of considerable interest. The centre of the group consists of scalar matrices with entries from the centre of the ring (cf. Centre of a ring) . When is commutative one defines the special linear group , which consists of matrices with determinant 1. When is a field, the commutator subgroup of the group coincides with (apart from the case , ), and any normal subgroup of is either contained in or contains . In particular, the projective special linear group

is a simple group (apart from the cases , ).

If is a skew-field and , any normal subgroup of is either contained in or contains the commutator subgroup of generated by transvections (cf. Transvection), and the quotient group is simple. Also, there exists a natural isomorphism

where is the multiplicative group of the skew-field . If is finite-dimensional over its centre , then the role of is played by the group of all matrices from with reduced norm 1. The groups and do not always coincide, although this is so if is a global field (see Kneser–Tits hypothesis).

The study of the normal structure of general linear groups over a ring is associated with algebraic -theory. The group over a general ring may contain numerous normal subgroups. For example, if is a commutative ring without zero divisors and with a finite number of generators, then is a residually-finite group, i.e. for each element there exists a normal subgroup of finite index not containing . In the case , the description of the normal subgroups of is in fact equivalent to the congruence problem for , since

and any non-scalar normal subgroup of the group for 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

[1]	E. Artin, "Geometric algebra" , Interscience (1957)
[2]	J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955)
[3]	H. Bass, "Algebraic -theory" , Benjamin (1968)