This paper gives a complete classification of the finite groups that contain a strongly closed p-subgroup for p any prime.

We consider splittings of groups over finite and two-ended subgroups. We study the combinatorics of such splittings using generalisations of Whitehead graphs. In the case of hyperbolic groups, we relate this to the topology of the boundary. In particular, we give a proof that the boundary of a one-ended strongly accessible hyperbolic group has no global cut point. AMS Classification 20F32

Given a finite group G and an element g∈G, we may compare the expected number e(G) of elements needed to generate e(G,g) together with g. We address following question: how large can difference e(G)−e(G,g) be? prove that in general this be arbitrarily large. For example for every positive integer n there exists 2-generated such e(G)≥n but e(G,g)≤5 some g∈G. However, if derived subgroup is nilpo...