# Sh:738

- Göbel, R., & Shelah, S. (2003).
*Philip Hall’s problem on non-abelian splitters*. Math. Proc. Cambridge Philos. Soc.,**134**(1), 23–31. arXiv: math/0009091 DOI: 10.1017/S0305004102006096 MR: 1937789 -
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. - 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}, doi = {10.1017/S0305004102006096}, mrclass = {20E22 (20J05)}, mrnumber = {1937789}, mrreviewer = {J. J. Woeppel}, 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} }