Element order versus minimal degree in permutation groups: an old lemma with new applications

TL;DR

This paper revisits and simplifies a key lemma about permutation order and support size, extending its applicability to various areas like algorithms, graph diameters, and combinatorial structures.

Contribution

It provides a cleaner, more general version of a 1987 lemma, removing unnecessary assumptions and broadening its application scope.

Findings

Simplified proof of the lemma

Broader applicability to permutation group problems

Enhanced understanding of permutation support and order

Abstract

In this note we present a simplified and slightly generalized version of a lemma the authors published in 1987. The lemma as stated here asserts that if the order of a permutation of $n$ elements is greater than $n^{\alpha}$ then some non-identity power of the permutation has support size less than $n/{\alpha}$. The original version made an unnecessary additional assumption on the cycle structure of the permutation; the proof of the present cleaner version follows the original proof verbatim. Application areas include parallel and sequential algorithms for permutation groups, the diameter of Cayley graphs of permutation groups, and the automorphisms of structures with regularity constraints such as Latin squares, Steiner 2-designs, and strongly regular graphs. This note also serves as a modest tribute to the junior author whose untimely passing is deeply mourned.