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

2025/05/13 by Fernando Andrés Benavides, Benavides, Fernando Andres, Wilson Fernando Mutis +1
Mathematics · Computer Science · Engineering · #Finite Group Theory Research #Coding theory and cryptography #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2505.08887

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=⟨ a,b : a^m=1,b²ⁿ=a^g,bab^-1=a^-1⟩. Consequently, we provide a new proof of the result for the dihedral group D_n and dicylic group Q_4n.

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