End ModP.
Section ModularGroupAction.
Variables (aT rT : finGroupType) (D : {group aT}) (R : {group rT}).
Variables (to : groupAction D R) (p : nat).
Implicit Types (G H : {group aT}) (M : {group rT}).

(* Specifications: *)
Lemma nontrivial_gacent_pgroup G M :
p.-group G -> p.-group M -> {acts G, on group M | to} ->
M :!=: 1 -> 'C_(M | to)(G) :!=: 1.
Lemma pcore_sub_astab_irr G M :
p.-group M -> M \subset R -> acts_irreducibly G M to ->
'O_p(G) \subset 'C_G(M | to).
Lemma pcore_faithful_irr_act G M :
p.-group M -> M \subset R -> acts_irreducibly G M to ->
[faithful G, on M | to] ->
'O_p(G) = 1.