proposition: Lemma Clifford_Socle1 : Socle sH = 1%:M.
proof: Proof. case/imsetP: Clifford_atrans => W _ _; have simW := socle_simple W. have [X sXG [def1 _]] := Clifford_basis simW. rewrite reducible_Socle1 //; apply: mxsemisimple_reducible. apply: intro_mxsemisimple def1 _ => x /(subsetP sXG) Gx _. exact: Clifford_simple. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path) mathcomp(div choice fintype tuple finfun bigop prime) mathcomp(ssralg poly polydiv finset fingroup morphism) mathcomp(perm automorphism quotient finalg action zmodp) mathcomp(commutator cyclic center pgroup matrix mxalgebra) mathcomp(mxpoly)