The Erd\H{o}s-Ginzburg-Ziv constant of rank-two-like $p$-groups

TL;DR

This paper refines bounds and determines exact values for the Erd ext{"o}s-Ginzburg-Ziv constant of certain $p$-groups, confirming a conjecture and expanding understanding of these combinatorial invariants.

Contribution

It provides the first exact values and improved bounds for the Erd ext{"o}s-Ginzburg-Ziv constant of rank-two-like $p$-groups, extending previous results and confirming Gao's conjecture.

Findings

Exact value of the Erd ext{"o}s-Ginzburg-Ziv constant for specific $p$-groups

Improved upper bounds for the constant in rank-two-like $p$-groups

Confirmation of Gao's conjecture for a new family of groups

Abstract

Adapting Reiher's proof of Kemnitz's conjecture, we obtain two refinements of a theorem of Schmid and Zhuang. Our main results provide improved upper bounds for the Erd\H{o}s-Ginzburg-Ziv constant of rank-two-like $p$-groups, and their direct products with cyclic groups of order coprime to $p$. In particular, we determine the exact value of this constant, and also confirm a conjecture of Gao, for a new infinite family of groups of arbitrarily large rank.