proposition: Lemma extend_cyclic_Mho G p x : p.-group G -> x \in G -> 'Mho^1(G) = <[x ^+ p]> -> forall k, k > 0 -> 'Mho^k(G) = <[x ^+ (p ^ k)]>.
proof: Proof. move=> pG Gx defG1 [//|k _]; have pX := mem_p_elt pG Gx. apply/eqP; rewrite eqEsubset cycle_subG (Mho_p_elt _ Gx pX) andbT. rewrite (MhoE _ pG) gen_subG; apply/subsetP=> ypk; case/imsetP=> y Gy ->{ypk}. have: y ^+ p \in <[x ^+ p]> by rewrite -defG1 (Mho_p_elt 1 _ (mem_p_elt pG Gy)). rewrite !expnS /= !expgM => /cycleP[j ->]. by rewrite -!expgM mulnCA mulnC expgM mem_cycle. 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)