End Socle.
Section Clifford.
Variables (gT : finGroupType) (G H : {group gT}).
Hypothesis nsHG : H <| G.
Variables (n : nat) (rG : mx_representation F G n).
Let sHG := normal_sub nsHG.
Let nHG := normal_norm nsHG.
Let rH := subg_repr rG sHG.

(* Specifications: *)
Lemma Clifford_simple M x : mxsimple rH M -> x \in G -> mxsimple rH (M *m rG x).
Lemma Clifford_hom x m (U : 'M_(m, n)) :
x \in 'C_G(H) -> (U <= dom_hom_mx rH (rG x))%MS.
Lemma Clifford_iso x U : x \in 'C_G(H) -> mx_iso rH U (U *m rG x).
Lemma Clifford_iso2 x U V :
mx_iso rH U V -> x \in G -> mx_iso rH (U *m rG x) (V *m rG x).
Lemma Clifford_componentJ M x :
mxsimple rH M -> x \in G ->
(component_mx rH (M *m rG x) :=: component_mx rH M *m rG x)%MS.