proposition: Lemma pnat_exponent pi G : pi.-nat (exponent G) = pi.-group G.
proof: Proof. congr (_ && _); first by rewrite cardG_gt0 exponent_gt0. apply: eq_all_r => p; rewrite !mem_primes cardG_gt0 exponent_gt0 /=. apply: andb_id2l => p_pr; apply/idP/idP=> pG. exact: dvdn_trans pG (exponent_dvdn G). by case/Cauchy: pG => // x Gx <-; apply: dvdn_exponent. 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)