The Erd\H{o}s-Ko-Rado property for some 2-transitive groups

TL;DR

This paper extends algebraic methods to determine the maximum intersecting sets in various 2-transitive groups, including Mathieu groups and small degree groups, generalizing known results for symmetric, alternating, and projective linear groups.

Contribution

It applies algebraic techniques to a broader class of 2-transitive groups, including Mathieu groups and small degree groups, to establish Erdős–Ko–Rado properties.

Findings

Maximum intersecting sets are cosets of point-stabilizers in many 2-transitive groups.

The method is successfully applied to Mathieu groups.

Results include all 2-transitive groups of degree up to 20.

Abstract

A subset of a group G of Sym(n) is intersecting if for any pair of permutations $\pi,\sigma \in G$ there is an $i$ in {1,2,...,n} such that $\pi(i) = \sigma(i)$. It has been shown, using an algebraic approach, that the largest intersecting sets in each of Sym(n), Alt(n) and PGL(2,q) are exactly the cosets of the point-stabilizers. In this paper, we show how this method can be applied more generally to many 2-transitive groups. We then apply this method to the Mathieu groups and to all 2-transtive groups with degree no more than 20.