On the maximal number of coprime subdegrees in finite primitive permutation groups

TL;DR

This paper proves that in any finite primitive permutation group, the maximum number of pairwise coprime non-trivial subdegrees is two, revealing a fundamental restriction on the structure of such groups.

Contribution

It establishes a universal upper bound of two on the size of sets of pairwise coprime non-trivial subdegrees in finite primitive groups, resolving a long-standing question.

Findings

Maximum of two pairwise coprime non-trivial subdegrees in finite primitive groups

Examples with two coprime subdegrees exist, but no larger sets are possible

Provides structural insights into the orbit sizes of primitive permutation groups

Abstract

The subdegrees of a transitive permutation group are the orbit lengths of a point stabilizer. For a finite primitive permutation group which is not cyclic of prime order, the largest subdegree shares a non-trivial common factor with each non-trivial subdegree. On the other hand it is possible for non-trivial subdegrees of primitive groups to be coprime, a famous example being the rank 5 action of the small Janko group on 266 points which has subdegrees of lengths 11 and 12. We prove that, for every finite primitive group, the maximal size of a set of pairwise coprime non-trivial subdegrees is at most 2.