Camina pairs that are not $p$-closed

TL;DR

This paper demonstrates that for each prime number, there exists a Camina pair where the subgroup is a p-group but the overall group is not p-closed, highlighting a specific structural property.

Contribution

It constructs examples of Camina pairs with p-group subgroups that are not p-closed, providing new insights into their structural properties.

Findings

Existence of Camina pairs with p-group N and non-p-closed G for all primes p

Counterexamples to p-closure in Camina pairs

Structural distinctions in Camina pairs

Abstract

We show for every prime $p$ that there exists a Camina pair $(G,N)$ where $N$ is a $p$-group and $G$ is not $p$-closed.