Fixpoint abelian_type_rec n G :=
if n is n'.+1 then if abelian G && (G :!=: 1) then
exponent G :: abelian_type_rec n' (tag (abelian_type_subproof G))
else [::] else [::].
Definition abelian_type (A : {set gT}) := abelian_type_rec #|A| <<A>>.

Theorem abelian_structure G :
abelian G ->
{b | \big[dprod/1]_(x <- b) <[x]> = G & map order b = abelian_type G}.