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

TL;DR

The paper addresses the minimal product-set problem in finite groups, denoting the minimal size as $\mu_G(r,s)$, and extends the known abelian-case link to the non-abelian metacyclic class $K_{m,n}=\langle a,b:\ a^m=1,\ b^{2n}=a^g,\ bab^{-1}=a^{-1}\rangle$. It develops a normal-subgroup parametrization via a bijection $\Psi$ on a parameter set $\Gamma$, and uses a double-induction framework to prove $\mu_G(r,s)=\kappa_G(r,s)$ for $G=K_{m,n}$ and $1\le r,s\le 2mn$, enabling the same equality for dihedral and dicyclic groups. Consequently, the paper yields new proofs of the equality for $D_n$ and $Q_{4n}$ and provides a unified approach for non-abelian minimal product-set problems. This advances understanding of product-set sizes in solvable, non-abelian groups and broadens the applicability of the $\mu_G=\kappa_G$ paradigm.

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 $K_{m,n}=\left\langle a,b \ : \ a^m=1,b^{2n}=a^g,bab^{-1}=a^{-1}\right\rangle$. Consequently, we provide a new proof of the result for the dihedral group $D_n$ and dicylic group $Q_{4n}$.

Theorems & Definitions (13)

• Definition 2.1
• Theorem 2.2: Eliahou and Kervaire Eliahou2010
• Theorem 2.3: Hölder
• Theorem 2.4: Yang Yang2020
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Proposition 3.3
• proof