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>>.

Lemma mul_card_Ohm_Mho_abelian n G :
abelian G -> (#|'Ohm_n(G)| * #|'Mho^n(G)|)%N = #|G|.