Some generalizations of Camina pairs and orders of elements in cosets

TL;DR

This paper explores generalizations of Camina pairs in finite groups, characterizing when certain conjugacy and induction conditions hold, and analyzing the structure of subgroups and elements with specific order properties.

Contribution

It establishes new criteria for generalized Camina pairs, links conjugacy conditions to subgroup structure, and characterizes groups with elements of odd order in cosets.

Findings

Irreducible characters induce homogeneously under specific conjugacy conditions.

Normal closure of subgroup is nilpotent if conjugacy conditions hold.

Structure of subgroup H determined when cosets of odd order elements contain only odd order elements.

Abstract

In this paper, we investigate certain generalizations of Camina pairs. Let $H$ be a nontrivial proper subgroup of a finite group $G$. We first show that every nontrivial irreducible complex character of $H$ induces homogeneously to $G$ if and only if for every $x\in G\setminus H$, the element $x$ is conjugate to $xh$ for all $h\in H$. Furthermore we prove that if $xh$ is conjugate to either $x$ or $x^{-1}$ for all $h\in H$ and all $x\in G\setminus H$, then the normal closure $N$ of $H$ in $G$ also satisfies the same condition, and $N$ is nilpotent. Finally, we determine the structure of $H$ under the assumption that for every element $x\in G\setminus H$ of odd order, the coset $xH$ consists entirely of elements of odd order.