proposition: Lemma exponent_dprod_homocyclic p K H G : K \x H = G -> p.-group G -> homocyclic G -> K :!=: 1 -> exponent K = exponent G.
proof: Proof. move=> defG pG homG ntK; have [homK _] := dprod_homocyclic defG pG homG. have [] := abelian_type_dprod_homocyclic defG pG homG. by rewrite abelian_type_homocyclic // -['r(K)]prednK ?rank_gt0 => [[]|]. Qed.
imports: mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path choice) mathcomp(div fintype finfun bigop finset prime binomial) mathcomp(fingroup morphism perm automorphism action) mathcomp(quotient gfunctor gproduct ssralg countalg finalg zmodp) mathcomp(cyclic pgroup gseries nilpotent sylow)