Philip Hall's problem on non-Abelian splitters

Abstract

Philip Hall raised around 1965 the following question which is stated in the Kourovka Notebook [11, p. 88]: 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 = 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 ω ∼ = G, Hom (L ω , S ω ) = 0 (S ω the infinite symmetric group acting on ω) and End (L, L) = Inn L ∪ {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 [8] which also shows that such groups exist, in fact there is a class of Hall-groups which is not a set.

Constructing Simple Groups for Localizations

“…Like [12], this shows (now in ZFC) that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg-Mac Lane space K(A, 1) for many groups A. The Main Theorem 1.3 is also used to answer a question by Philip Hall in [13]. …”

“…The Main Theorem 1.3 will also be used to answer a problem of Philip Hall from 1966 in [13] mentioned in the Kourovka notebook. There is a class of groups G such that any extension of G by a copy of G is isomorphic to G. Only some of the properties of the groups in our Main Theorem 1.3 will only be used for the Hall problem.…”