Philip Hall's Problem On Non-Abelian Splitters

Abstract

Philip Hall raised around 1965 the following question which is stated in the Kourovka Notebook: Is there a non-trivial group which is isomorphic with every proper extension of itself by itself? We will decompose the problem into two parts: We want to find non-commutative splitters, that are groups G not= 1 with Ext(G,G)=1 . The class of splitters fortunately is quite large so that extra properties can be added to G. We can consider groups G with the following properties: There is a complete group L with cartesian product L^omega cong G, Hom(L^omega,S_omega)=0 (S_omega the infinite symmetric group acting on omega) and End(L,L)=Inn(L) cup {0}. We will show that these properties ensure that G is a splitter and hence obviously a Hall-group in the above sense. Then we will apply a recent result from our joint paper math.GR/0009089 which also shows that such groups exist, in fact there is a class of Hall-groups which is not a set.