proposition: Lemma Mho_p_cycle p x : p.-elt x -> 'Mho^n(<[x]>) = <[x ^+ (p ^ n)]>.
proof: Proof. move=> p_x. apply/eqP; rewrite (MhoE p_x) eqEsubset cycle_subG mem_gen; last first. by apply: imset_f; apply: cycle_id. rewrite gen_subG andbT; apply/subsetP=> _ /imsetP[_ /cycleP[k ->] ->]. by rewrite -expgM 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)