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

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 $\mathrm{\mid }A\mathrm{\mid }=r$ and $\mathrm{\mid }B\mathrm{\mid }=s$. 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 $K_{m,n}=⟨a,b:a^m=1,b^{2n}=a^g,ba{b}^{-1}=a^{-1}⟩$. Consequently, we provide a new proof of the result for the dihedral group $D_n$ and dicylic group $Q_{4n}$.