Factorizing a Finite Group into Conjugates of a Subgroup

TL;DR

This paper investigates how finite groups can be expressed as products of conjugates of a subgroup, establishing bounds on the minimal number needed based on the group's solvability and size.

Contribution

It introduces the invariant mma_{cp}(G) and provides bounds for non-solvable and solvable groups, advancing understanding of group factorizations.

Findings

mma_{cp}(G)  36 for non-solvable groups

mma_{cp}(G) can be any integer > 2 for solvable groups

mma_{cp}(G)  4 \, \, 2\log_2|G| for general groups

Abstract

For every non-nilpotent finite group $G$, there exists at least one proper subgroup $M$ such that $G$ is the setwise product of a finite number of conjugates of $M$. We define $\gamma_{\text{cp}}\left( G\right) $ to be the smallest number $k$ such that $G$ is a product, in some order, of $k$ pairwise conjugated proper subgroups of $G$. We prove that if $G$ is non-solvable then $\gamma_{\text{cp}}\left( G\right) \leq36$ while if $G$ is solvable then $\gamma_{\text{cp}}\left( G\right) $ can attain any integer value bigger than $2$, while, on the other hand, $\gamma_{\text{cp}}\left( G\right) \leq4\log_{2}\left\vert G\right\vert $.