# Sh:738

• Göbel, R., & Shelah, S. (2003). Philip Hall’s problem on non-abelian splitters. Math. Proc. Cambridge Philos. Soc., 134(1), 23–31.
• 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\neq 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^\omega\cong G, {\rm Hom}(L^\omega, S_\omega)=0 (S_\omega the infinite symmetric group acting on \omega) 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 [GbSh:739] which also shows that such groups exist, in fact there is a class of Hall-groups which is not a set.
• Version 2000-08-28_11 (11p) published version (9p)
Bib entry
@article{Sh:738,
author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon},
title = {{Philip Hall's problem on non-abelian splitters}},
journal = {Math. Proc. Cambridge Philos. Soc.},
fjournal = {Mathematical Proceedings of the Cambridge Philosophical Society},
volume = {134},
number = {1},
year = {2003},
pages = {23--31},
issn = {0305-0041},
mrnumber = {1937789},
mrclass = {20E22 (20J05)},
doi = {10.1017/S0305004102006096},
note = {\href{https://arxiv.org/abs/math/0009091}{arXiv: math/0009091}},
arxiv_number = {math/0009091},
keyword = {self-splitting non-commutative groups, infinite simple groups}
}