Centralizers of normal subgroups and the Z⁎Z⁎-theorem

Glauberman's Z⁎Z⁎-theorem and analogous statements for odd primes show that, for any prime p and any finite group G with Sylow p-subgroup S  , the centre of G/Op′(G)G/Op′(G) is determined by the fusion system FS(G)FS(G). Building on these results we show a statement that seems a priori more general: For any normal subgroup H of G   with Op′(H)=1Op′(H)=1, the centralizer CS(H)CS(H) is expressed in terms of the fusion system FS(G)FS(G) and its normal subsystem induced by H.