Factorizing a Finite Group into Conjugates of a Subgroup

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 $.