Section RawGroupAction.
Variables (aT rT : finGroupType) (D : {set aT}) (R : {set rT}).
Variable to : groupAction D R.
Canonical gacent_group A := Group (group_set_gacent A).
Canonical actby_groupAction := GroupAction actby_is_groupAction.

Lemma gacent_actby B :
'C_(|actby_groupAction)(B) = 'C_(G | to)(A :&: B).