Minimal Product Set in Non-Abelian Metacyclic Groups of Even Order

Abstract

Given a finite group $G$ and positive integers $r$ and $s$, a problem of interest in algebra is determining the minimum cardinality of the product set $AB$, where $A$ and $B$ are subsets of $G$ such that $|A|=r$ and $|B|=s$. This problem has been solved for the class of abelian groups; however, it remains open for finite non-abelian groups. In this paper, we prove that the result obtained for abelian groups can be extended to the class of metacyclic groups $K_{m,n}=\left\langle a,b \ : \ a^m=1,b^{2n}=a^g,bab^{-1}=a^{-1}\right\rangle$. Consequently, we provide a new proof of the result for the dihedral group $D_n$ and dicylic group $Q_{4n}$.