Editing Special linear group (section)

==Relations to other subgroups of GL(''n'', ''A'')==
{{see also|Whitehead's lemma}}

Two related subgroups, which in some cases coincide with <math>\operatorname{SL}</math>, and in other cases are accidentally conflated with <math>\operatorname{SL}</math>, are the [[commutator subgroup]] of <math>\operatorname{GL}</math>, and the group generated by [[Shear mapping|transvections]]. These are both subgroups of <math>\operatorname{SL}</math> (transvections have determinant 1, and det is a map to an abelian group, so <math>[\operatorname{GL},\operatorname{GL}]<\operatorname{SL}</math>), but in general do not coincide with it.

The group generated by transvections is denoted <math>\operatorname{E}(n,A)</math> (for [[elementary matrices]]) or <math>\operatorname{TV}(n,A)</math>. By the second [[Steinberg relations|Steinberg relation]], for <math>n\geq 3</math>, transvections are commutators, so for <math>n\geq 3</math>, <math>\operatorname{E}(n,A)<[\operatorname{GL}(n,A),\operatorname{GL}(n,A)]</math>.

<!-- Does equality hold? Dunno. -->
For <math>n=2</math>, transvections need not be commutators (of <math>2\times 2</math> matrices), as seen for example when <math>A</math> is <math>\mathbb{F}_2</math>, the field of two elements. In that case 
:<math>A_3 \cong [\operatorname{GL}(2, \mathbb{F}_2),\operatorname{GL}(2, \mathbb{F}_2)] < \operatorname{E}(2, \mathbb{F}_2) = \operatorname{SL}(2, \mathbb{F}_2) = \operatorname{GL}(2, \mathbb{F}_2) \cong S_3,</math>
where <math>A_3</math> and <math>S_3</math> respectively denote the [[alternating group|alternating]] and [[symmetric group]] on 3 letters.

However, if <math>A</math> is a field with more than 2 elements, then {{nowrap|1=E(2, ''A'') = [GL(2, ''A''), GL(2, ''A'')]}}, and if <math>A</math> is a field with more than 3 elements, {{nowrap|1=E(2, ''A'') = [SL(2, ''A''), SL(2, ''A'')]}}. {{Dubious - discuss|date=March 2019}}

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]] <math>SK_1(A)=\operatorname{SL}(A)/\operatorname{E}(A)</math>, where <math>\operatorname{SL}(A)</math> and <math>\operatorname{E}(A)</math> are the [[direct limit of groups|stable group]]s of the special linear group and elementary matrices.