Relations to other subgroups of GL.28n.2CA.29 Special linear group
two related subgroups, in cases coincide sl, , in other cases accidentally conflated sl, commutator subgroup of gl, , group generated transvections. these both subgroups of sl (transvections have determinant 1, , det map abelian group, [gl, gl] ≤ sl), in general not coincide it.
the group generated transvections denoted e(n, a) (for elementary matrices) or tv(n, a). second steinberg relation, n ≥ 3, transvections commutators, n ≥ 3, e(n, a) ≤ [gl(n, a), gl(n, a)].
for n = 2, transvections need not commutators (of 2 × 2 matrices), seen example when f2, field of 2 elements, then
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{\displaystyle \operatorname {alt} (3)\cong [\operatorname {gl} (2,\mathbf {f} _{2}),\operatorname {gl} (2,\mathbf {f} _{2})]<\operatorname {e} (2,\mathbf {f} _{2})=\operatorname {sl} (2,\mathbf {f} _{2})=\operatorname {gl} (2,\mathbf {f} _{2})\cong \operatorname {sym} (3),}
where alt(3) , sym(3) denote alternating resp. symmetric group on 3 letters.
however, if field more 2 elements, e(2, a) = [gl(2, a), gl(2, a)], , if field more 3 elements, e(2, a) = [sl(2, a), sl(2, a)].
in circumstances these coincide: special linear group on field or euclidean domain generated transvections, , stable special linear group on dedekind domain generated transvections. more general rings stable difference measured special whitehead group sk1(a) := sl(a)/e(a), sl(a) , e(a) stable groups of special linear group , elementary matrices.
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