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October 9, 2025Open Access

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

FBFernando Andrés BenavidesUniversity of Nariño WMWilson Fernando MutisUniversity of Nariño

Key Points

The minimum cardinality of the product set can be determined in metacyclic groups, expanding previous results.
For metacyclic groups, we establish a new proof of results applicable to both the dihedral group and the dicyclic group.
This study resolves an open problem for non-abelian groups by adapting methods from abelian group theory.
The results indicate a deeper understanding of the structure of non-abelian groups and their product sets.

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₌, ₍= a, b \: \ aᵐ=1, b^2n=aᵍ, bab^-1=a^-1. Consequently, we provide a new proof of the result for the dihedral group Dₙ and dicylic group Q₄₍.

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