The Uniqueness of the Small Mathieu Groups Revisited

All known finite sharply 4-transitive permutation sets containing the identity are groups, namely , , and the Mathieu group of degree 11.

In a joint work with P. Quattrocchi it is proved that a sharply 4-transitive permutation SET on 11 elements containing the identity must necessarily be the Mathieu group. The proof uses direct counting arguments. It is based on a combinatorial property of the involutions in the Mathieu group (which is established here) and on the uniqueness of the Minkowski planes of order 9 (which had been established before): the validity of both facts relies on computer calculations. A similar property holds for the Mathieu group of degree 12, when `sharply 4-transitive' is replaced by `sharply 5-transitive'. There is also some connection with J. Conway's .

The above result has the following consequence. Let G be a finite sharply 4-transitive permutation set containing the identity with the property that if a permutation g lies in G then so does its inverse (in the geometric structure arising from G it means that at least one block-symmetry is an automorphism): then G is forced to be one of the four known examples.