Semisimple groups acting on semisimple groups

Suppose A and B are finite semisimple groups, i.e. direct products of simple nonabelian groups. Assume A and B are subgroups of a group G such that B is normalized by A and A not less than or equal to B. It is shown that then the centralizer C-AB (B) is not trivial provided P(A) >= P(B), where P(X) denotes the smallest degree of a faithful permutation representation of a group X. Hence B cannot be the generalized Fitting subgroup or the socle of its normalizer in G. Further consequences are derived.