f.
apply: is_total_action => [G | G x y]; apply: val_inj; rewrite /= ?act1 //.
exact: actM.Notation "'J" := (@conjg_action _) (at level 8) : action_scope.

(* Specifications: *)
Lemma index_cent1 x : #|G : 'C_G[x]| = #|x ^: G|.
Definition autact := act ('P \ subsetT (Aut
Lemma autactK a : actperm aut_action a = a.