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Infinite sharply multiply transitive groups. (English) Zbl 1345.20003

This paper is a survey on sharply \(n\)-transitive permutation groups (\(n\geq 2\)).
From the summary: “The finite sharply 2-transitive groups were classified by Zassenhaus in the 1930’s. They essentially all look like the group of affine linear transformations \(x\mapsto ax+b\) for some field (or at least near-field) \(K\). However, the question remained open whether the same is true for infinite sharply 2-transitive groups. There has been extensive work on the structures associated to such groups indicating that Zassenhaus’ results might extend to the infinite setting. For many specific classes of groups, like Lie groups, linear groups, or groups definable in \(o\)-minimal structures it was indeed proved that all examples inside the given class arise in this way as affine groups. However, it recently turned out that the reason for the lack of a general proof was the fact that there are plenty of sharply 2-transitive groups which do not arise from fields or near-fields! In fact, it is not too hard to construct concrete example”.

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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References:

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