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 ${\gamma }_{\text{cp}}\left(G\right)\le 36$ 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, ${\gamma }_{\text{cp}}\left(G\right)\le 4{\log }_2\left|G\right|$.

14 pages

References in corpus (2)

• Growth in linear groups
• Unitriangular factorisations of Chevalley groups