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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.08887 |
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Table of Contents:
- 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 $Q_{4n}$.