Weak Freiman isomorphisms and sequencings of small sets

TL;DR

This paper introduces a weakened form of Freiman isomorphisms and demonstrates their application in proving sequenceability of small subsets in various non-abelian groups, with bounds related to prime factors of group parameters.

Contribution

It develops a new concept of weak Freiman isomorphisms and applies it to establish sequenceability results for subsets of dihedral and dicyclic groups.

Findings

Subsets of dihedral groups are sequenceable if prime factors of m are larger than k!

Refined bound of k!/2 for cyclic groups improves previous results

Subsets of dicyclic groups are sequenceable if prime factors of m are larger than k^k

Abstract

In this paper, we introduce a weakening of the Freiman isomorphisms between subsets of non necessarily abelian groups. Inspired by the breakthrough result of Kravitz, [14], on cyclic groups, as a first application, we prove that any subset of size $k$ of the dihedral group $D_{2m}$ (and, more in general, of a class of semidirect products) is sequenceable, provided that the prime factors of $m$ are larger than $k!$. Also, a refined bound of $k!/2$ for the size of the prime factors of $m$ can be obtained for cyclic groups $\mathbb{Z}_m$, slightly improving the result of [14]. Then, applying again the concept of weak Freiman isomorphism, we show that any subset of size $k$ of the dicyclic group $\mathrm{Dic}_{m}$ is sequenceable, provided that the prime factors of $m$ are larger than $k^k$.