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Philip Hall's problem on non-Abelian splitters

Published online by Cambridge University Press:  12 March 2003

RÜDIGER GÖBEL
Affiliation:
Fachbereich 6, Mathematik und Informatik Universität Essen, 45117 Essen, Germany. e-mail: R.Goebel@Uni-Essen.De
SAHARON SHELAH
Affiliation:
Department of Mathematics Hebrew University, Jerusalem, Israel. e-mail: Shelah@math.huji.ac.il Alternative address for Saharon Shelah: Rutgers University, New Brunswick, NJ, U.S.A.

Abstract

Philip Hall raised the following question which is stated in The Kourovka Notebook [12, p. 88]: is there a non-trivial group which is isomorphic with every proper extension of itself by itself? We will split the problem into two parts: we want to find non-commutative splitters, that are groups $G\ne 1$ with ${\rm Ext} (G, G) = 1$. 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 $L^w \cong G, {\rm Hom}(L^w, S^w) = 0$($S_w$ the infinite symmetric group acting on $w$) and ${\rm End} (L, L) = {\rm Inn}\, L\cup\{0\}$. 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 [9] which also shows that such groups exist; in fact there is a class of Hall groups which is not a set.

Type
Research Article
Copyright
2003 Cambridge Philosophical Society

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