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Lemma 74 ([Taking cores without losing a simple image]). Let R_0≤D , let C R_0 , and suppose that R_0/C is nonabelian simple. Let L_1,…,L_s≤D be normalized by R_0 . Suppose that, for every t , there is a subgroup D_t R_0 such that D_tC=R_0, D_t^ L_t =D_t, [D_t,L_t]≤D_t∩C. Then, for W=⟨R_0,L_1,…,L_s⟩ , the subgroup core_W(R_0) maps onto R_0/C . No normalization of C by L_t is assumed.