2000/09/08 by Rüdiger Göbel, Saharon Shelah, Göbel, Rüdiger +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.GR #math.LO #math.RA
paper · pdf · doi:10.48550/arxiv.math/0009091
published as Math. Proc. Cambridge Philos. Soc. 134 No. 1 (2003) 23--31
arxiv created 2000/09/08 · openalex publication_date 2000/09/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Lomega cong G, Hom(Lomega,Somega)=0 (Somega 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.