proposition: Lemma Ohm1Eexponent p G : prime p -> exponent 'Ohm_1(G) %| p -> 'Ohm_1(G) = 'Ldiv_p(G).
proof: Proof. move=> p_pr expG1p; have pG: p.-group G. apply: sub_in_pnat (pnat_pi (cardG_gt0 G)) => q _. rewrite -piOhm1 mem_primes; case/and3P=> q_pr _; apply: pgroupP q_pr. by rewrite -pnat_exponent (pnat_dvd expG1p) ?pnat_id. apply/eqP; rewrite eqEsubset {2}(OhmE 1 pG) subset_gen subsetI Ohm_sub. by rewrite sub_LdivT expG1p. 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)