Products of subgroups, subnormality, and relative orders of elements

TL;DR

This paper characterizes elements in a group based on their powers relative to subgroups of finite index and explores conditions under which subgroup products have orders dividing certain group indices.

Contribution

It provides an explicit description of elements satisfying specific power conditions related to finite index subgroups and investigates divisibility properties of subgroup products.

Findings

Explicit description of elements with $x^{|G:H|} \

Conditions for divisibility of subgroup product orders by group indices

Connections between element powers and subgroup normality

Abstract

Let $G$ be a group. We give an explicit description of the set of elements $x \in G$ such that $x^{|G:H|} \in H$ for every subgroup of finite index $H \leqslant G$. This is related to the following problem: given two subgroups $H$ and $K$, with $H$ of finite index, when does $|HK:H|$ divide $|G:H|$?