Weak Freiman isomorphisms and sequencings of small sets

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 m (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 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 Dic m is sequenceable, provided that the prime factors of m are larger than k k .