SUM AND DIFFERENCE SETS IN GENERALIZED QUATERNION GROUPS

Abstract

Given a group G, we say that a set [Formula presented] has more sums than differences (MSTD) if |A+A| > |A – A|, has more differences than sums (MDTS) if |A+A| < |A–A|, or is balanced if |A+A| = |A–A|. A problem of recent interest has been to understand the frequencies of these types of subsets. It is known that for arbitrary finite groups G, almost all subsets [Formula presented] are balanced sets as [Formula presented]. Recently for the generalized dihedral groups [Formula presented], it is conjectured that there are more MSTD sets than MDTS sets. In this paper, we investigate the behavior of the sum and difference sets of [Formula presented], where Q<inf>4n</inf> denotes generalized quaternion groups and show that the generalized quaternion group Q<inf>4n</inf> has at least 22n subsets which are MSTD. We also analyze the expectation for |A – A| where [Formula presented], proving an explicit formula for |A – A| when n is prime. © 2024 Jangjeon Research Institute for Mathematical Sciences and Physics. All rights reserved.