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Sharply 2-transitive groups. (English) Zbl 1343.20002

The finite sharply \(2\)-transitive groups were classified by H. Zassenhaus in the 1930’s, they are Frobenius groups of the form \(G=NH\) with \(N\) normal and such that \(|H|=|N|-1\). The last equality implies that in \(N\) there is a unique conjugacy class of non trivial elements and this forces \(N\) to be an elementary abelian \(p\)-group for some prime \(p\).
It remained an open question whether the infinite sharply 2-transitive groups admit an abelian normal subgroup. The first counterexamples were recently constructed by E. Rips, Y. Segev and K. Tent [“A sharply 2-transitive group without a non-trivial abelian normal subgroup” arXiv:1406.0382 (2014)]; in these examples involutions have no fixed points.
In the paper under review the authors give an alternative approach to such a construction by using partially defined group actions. As a corollary they prove that any group can be extended to a group acting sharply 2-transitively on some appropriate set without nontrivial abelian normal subgroup.

MSC:

20B22 Multiply transitive infinite groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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