Philip Hall's problem on non-Abelian splitters

by Goebel and Shelah. [GbSh:738]
Math. Proc. Camb. Phil. Soc., 2003
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 L^omega cong G, Hom (L^omega, S_omega)=0 (S_omega 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 [GbSh:739] which also shows that such groups exist, in fact there is a class of Hall-groups which is not a set.


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