Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

TL;DR

This paper investigates how finite groups can be expressed as products of conjugate subgroups that are either solvable or nilpotent, providing bounds on the minimal number of factors needed and applying structure theory of finite simple groups.

Contribution

It offers new bounds on the minimal length of conjugate factorizations by solvable or nilpotent subgroups in finite groups, utilizing the structure of finite simple groups and Carter subgroups.

Findings

Every finite simple group of Lie type is a product of four or three unipotent Sylow subgroups.

Derived an upper bound on the minimal length of solvable conjugate factorizations.

Provided an upper bound on the minimal length of nilpotent conjugate factorizations in finite groups.

Abstract

We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of finite simple groups of Lie type to give a unified self-contained proof that every such group is a product of four or three unipotent Sylow subgroups. Then we derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group. Finally, using conjugate factorizations of a general finite solvable group by any of its Carter subgroups, we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.