A sharply 2-transitive group without a non-trivial abelian normal subgroup

TL;DR

This paper constructs sharply 2-transitive groups that lack non-trivial abelian normal subgroups, answering a longstanding open question in group theory.

Contribution

It demonstrates the existence of such groups, expanding understanding of the structure of sharply 2-transitive groups.

Findings

Constructed examples of sharply 2-transitive groups without abelian normal subgroups

Showed any group can be embedded into such a group

Involutions in these groups have no fixed points

Abstract

We show that any group $G$ is contained in some sharply 2-transitive group $\mathcal{G}$ without a non-trivial abelian normal subgroup. This answers a long-standing open question. The involutions in the groups $\mathcal{G}$ that we construct have no fixed points.