### 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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