p-Group Camina pairs

TL;DR

This paper proves that if (G,Z(G)) is a Camina pair, then G is a p-group and establishes bounds relating the size of the center to the index of the center, exploring possible constructions beyond these bounds.

Contribution

It demonstrates that G must be a p-group for Camina pairs and provides bounds on the size of Z(G) relative to G, advancing understanding of Camina pair structures.

Findings

G is a p-group for Camina pairs

Established bound: |Z(G)| < |G:Z(G)|^{3/4}

Discussed potential for larger centers beyond current bounds

Abstract

Let $(G,Z(G))$ be a Camina pair. We prove that $G$ must be a $p$-group for some prime $p$. We also prove that $|Z(G)| < |G:Z(G)|^{3/4}$. Also, we discuss how one might build examples with $|Z(G)| > |G:Z(G)|^{1/2}$, although we are not able to prove the existence of such examples.