Baumann-components of finite groups of characteristic p, the W(B)-theorem

This paper completes the investigation of finite CK-groups of characteristic p in terms of their Baumann components we began in [Baumann-components of finite groups in characteristic p, general theory] and [Baumann-components of finite groups in characteristic p, reduction theorems]. In this paper we define for each finite p-group B a non-trivial characteristic subgroup W(B) and for each finite CK-group G of characteristic p with B in Baup(G), subnormal subgroups of G called Baumann blocks of G. We prove that G = N_G(W(B))E_W(G), where E_W(G) is the normal subgroup generated by the Baumann blocks of G. Moreover, we give the exact structure of the Baumann blocks of G and show that any two distinct Baumann blocks centralize each other.