Fact mx_repr_is_groupAction : is_groupAction [set: 'rV[R]_n] mx_repr_action.
Canonical Structure mx_repr_groupAction := GroupAction mx_repr_is_groupAction.
End FinRingRepr.
Notation "''MR' rG" := (mx_repr_action rG)
(at level 10, rG at level 8) : action_scope.
Notation "''MR' rG" := (mx_repr_groupAction rG) : groupAction_scope.
Section FinFieldRepr.
Variable F : finFieldType.
(* The external group action (by scaling) of the multiplicative unit group   *)
(* of the finite field, and the correspondence between additive subgroups    *)
(* of row vectors that are stable by this action, and the matrix row spaces. *)
Section ScaleAction.
Variables m n : nat.
Definition scale_act (A : 'M[F]_(m, n)) (a : {unit F}) := val a *: A.

Lemma scale_actE A a : scale_act A a = val a *: A. Proof. by []. Qed.
Fact scale_is_action : is_action setT scale_act.