Group cosets with all elements of equal order

TL;DR

This paper introduces equal order pairs $(G,N)$, where $G$ is finite and $N\lhd G$ is nontrivial, such that every element outside $N$ has the same order in its coset $xN$ as $x$ itself, generalizing Camina pairs. It adapts Camina-pair techniques to derive structural constraints, differentiating the nilpotent and non-nilpotent cases for $N$ and exploring consequences for the Fitting subgroup, Sylow subgroups, and quotient structures; it also provides extensive constructions of equal order pairs via direct products, Frobenius-type extensions, and semidirect products, yielding both solvable and nonsolvable examples. The main results, encapsulated in Theorems A–D, give precise restrictions on the position of $N$ relative to ${\bf F}(G)$, describe the Sylow-p-subgroup structure of $G/N$ under nilpotence of $N$, and show that when $N$ is not nilpotent, $G/N$ is a $p$-group with substantial further implications for solvable and nonsolvable cases, including an explicit minimal nonsolvable example related to ${\rm M}_{10}$. These findings extend Camina-pair theory, connect to $p'$-semiregularity, and reveal new families of equal order pairs beyond Frobenius groups, with several open questions about the Fitting height of $N$ in this context.

Abstract

Let $G$ be a finite group and $N$ a proper, nontrivial, normal subgroup of $G$. If, for every element $x$ of $G$ not lying in $N$, the elements in the coset $xN$ all have the same order as $x$, then we say that $(G,N)$ is an {\it{equal order pair}}. This generalizes the concept of a Camina pair, that was introduced by the first author. In the present paper we study several properties of equal order pairs, showing that in many respects they resemble Camina pairs, but with some important differences.

Theorems & Definitions (38)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.3
• Proposition 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6