Section MorphGroupAction.
Variables (aT1 aT2 rT1 rT2 : finGroupType).
Variables (D1 : {group aT1}) (D2 : {group aT2}).
Variables (R1 : {group rT1}) (R2 : {group rT2}).
Variables (to1 : groupAction D1 R1) (to2 : groupAction D2 R2).
Variables (h : {morphism R1 >-> rT2}) (f : {morphism D1 >-> aT2}).
Hypotheses (iso_h : isom R1 R2 h) (iso_f : isom D1 D2 f).
Hypothesis hfJ : {in R1 & D1, morph_act to1 to2 h f}.
Implicit Types (A : {set aT1}) (S : {set rT1}) (M : {group rT1}).

(* Specifications: *)
Lemma morph_gastabs S : S \subset R1 -> f @* 'N(S | to1) = 'N(h @* S | to2).
Lemma morph_gastab S : S \subset R1 -> f @* 'C(S | to1) = 'C(h @* S | to2).
Lemma morph_gacent A : A \subset D1 -> h @* 'C_(|to1)(A) = 'C_(|to2)(f @* A).
Lemma morph_gact_irr A M :
A \subset D1 -> M \subset R1 ->
acts_irreducibly (f @* A) (h @* M) to2 = acts_irreducibly A M to1.