Special Linear Group - Relations To Other Subgroups of GL(n,A)

Relations To Other Subgroups of GL(n,A)

Two related subgroups, which in some cases coincide with SL, and in other cases are accidentally conflated with SL, are the commutator subgroup of GL, and the group generated by transvections. These are both subgroups of SL (transvections have determinant 1, and det is a map to an abelian group, so ≤ SL), but in general do not coincide with it.

The group generated by transvections is denoted E(n, A) (for elementary matrices) or TV(n, A). By the second Steinberg relation, for n ≥ 3, transvections are commutators, so for n ≥ 3, E(n, A) ≤ .

For n = 2, transvections need not be commutators (of 2×2 matrices), as seen for example when A is F₂, the field of two elements, then

where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.

However, if A is a field with more than 2 elements, then E(2, A) =, and if A is a field with more than 3 elements, E(2, A) = .

In some circumstances these coincide: the special linear group over a field or a Euclidean domain is generated by transvections, and the stable special linear group over a Dedekind domain is generated by transvections. For more general rings the stable difference is measured by the special Whitehead group SK₁(A) := SL(A)/E(A), where SL(A) and E(A) are the stable groups of the special linear group and elementary matrices.