Weak Freiman isomorphisms and sequencings of small sets

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 $k^k$.