proposition: Lemma pnElem0 p G : 'E_p^0(G) = [set 1%G].
proof: Proof. apply/setP=> E; rewrite !inE -andbA; apply/and3P/idP=> [[_ pE] | /eqP->]. apply: contraLR; case/(pgroup_pdiv (abelem_pgroup pE)) => p_pr _ [k ->]. by rewrite pfactorK. by rewrite sub1G abelem1 cards1 logn1. 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)