=> Gx; suffices: x \in rker (irr_repr 1) by case/rkerP.
apply: subsetP x Gx; rewrite rker_submod rfix_mx_rstabC // -irr1_rfix.
by apply: component_mx_id; apply: socle_simp

& forall x, x \in G -> rG1 x *m B = B *m rG2 x.