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A sharply 2-transitive group without a non-trivial abelian normal subgroup. (English) Zbl 1483.20002

From the text: 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.
The finite sharply 2-transitive groups were classified by H. Zassenhaus in [Abh. Math. Semin. Hamb. Univ. 11, 187–220 (1935; Zbl 0011.10302)] and it is known that any finite sharply 2-transitive group contains a non-trivial abelian normal subgroup. In the infinite situation no classification is known. It was a long standing open problem whether every infinite sharply 2-transitive group contains a non-trivial abelian normal subgroup.
In [Comment. Math. Helv. 26, 203-224 (1952; Zbl 0047.26002)] J. Tits proved that this holds for locally compact connected sharply 2-transitive groups. Several other papers showed that under certain special conditions the assertion holds. The reader may wish to consult Appendix A for more detail, and for a description of our main results using permutation group theoretic language.
An equivalent formulation to the above problem is whether every near-domain is a near-field. We here show that this is not the case. We construct a sharply 2-transitive infinite group without a non-trivial abelian normal subgroup. In fact, the construction is similar in flavor to the free completion of partial generalized polgyons [K. Tent, Pure Appl. Math. Q. 7, No. 3, 1037–1052 (2011; Zbl 1232.51009)].

MSC:

20B22 Multiply transitive infinite groups
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