Camina triples

TL;DR

This paper explores the properties of Camina triples, a generalization of Camina pairs, revealing structural conditions such as nilpotency and p-group characteristics in related groups.

Contribution

It introduces the concept of Camina triples and establishes key structural results, extending the theory of Camina pairs in group theory.

Findings

If (G,N,M) is a Camina triple, then G/N is a p-group.

In a Camina triple, M is nilpotent.

M has a non-trivial nilpotent quotient.

Abstract

In this paper, we study Camina triples. Camina triples are a generalization of Camina pairs. Camina pairs were first introduced in 1978 by A.R. Camina in \cite{camina1}. Camina's work in \cite{camina1} was inspired by the study of Frobenius groups. We show that if $(G,N,M)$ is a Camina triple, then either $G/N$ is a $p$-group, $M$ is nilpotent, or $M$ has a non-trivial nilpotent quotient.