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 $|A|=r$ 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 Km,n= a,b \ : \ aᵐ=1,b²ⁿ=aᵍ,bab⁻¹=a⁻¹. Consequently, we provide a new proof of the result for the dihedral group Dₙ and dicylic group Q₄ₙ.