turkmath.org

A typical example of a sharply 2-transitive action is the action of the group of affine transformations of a field. In fact, for this example, we do not need all the field axioms, working over a “near field” is sufficient. This is the so-called standard sharply 2-transitive group. In 1936, Zassenhaus proved that all finite sharply 2-transitive groups are standard. In 1952, Tits extended this result to locally compact connected groups. Later, more results from different authors came verifying that under certain assumptions a sharply 2-transitive group is standard. However, the general problem of whether any sharply 2-transitive group is standard or not, remained open until 2014, when Rips, Segev and Tent, constructed the first non-standard example of a sharply 2-transitive group, and answered the almost a century-old problem negatively. Quickly after that, again in 2014, Tent and Ziegler constructed a much easier example. However, all these examples are of permutation characteristic 2 and none of them is of finite Morley rank. Therefore, there are still many open questions related to the subject. In my talk, I am planning to review the history and fundamental results in the area, then talk about the Tent-Ziegler example, and finally mention some old and new results in the finite Morley rank context.