The Erdős-Ginzburg-Ziv constant of rank-two-like $p$-groups

TL;DR

This work advances the study of the Erdős-Ginzburg-Ziv constant for rank-two-like $p$-groups by adapting Reiher's approach via the Baker–Schmidt theorem. It obtains a sharpened upper bound $\mathsf{s}(G)\le \mathsf{D}(G)+2\exp(G)-p^k-1$ whenever $\mathsf{D}(G)\le 2\exp(G)-p^k$, and proves exact values in the case $\mathsf{D}(G)=2\exp(G)-p^k$, thereby confirming Gao's conjecture for a new infinite family. A second bound $\mathsf{s}(G)\le \mathsf{D}(G)+2\exp(G)-(\tfrac{c-1}{2})-2$ is established when $\mathsf{D}(G)=2\exp(G)-c$, refining results for higher-rank $p$-groups. The authors extend these results to direct products with cyclic groups coprime to $p$ and derive a range of corollaries, including monotonicity results for $\mathsf{s}_{[j,n]}(G)$ and implications for Luo's conjectures. Overall, the paper deepens understanding of zero-sum invariants in high-rank $p$-groups and settles Gao-type conjectures for broad new families.

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 Erdős-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.

Theorems & Definitions (40)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Corollary 2.2
• Corollary 2.3
• Theorem 2.4
• Corollary 2.5