Sylow numbers and the structure of finite groups

TL;DR

This paper investigates how Sylow numbers influence the structure of finite groups, providing new bounds on various group lengths and extending previous results in the field.

Contribution

It introduces new bounds on p-length, nilpotent length, and derived length of finite groups using Sylow numbers, extending earlier work by Zhang.

Findings

Derived new bounds for p-length, nilpotent length, and derived length.

Extended known results of Zhang (1995).

Applied Sylow numbers to analyze group structure.

Abstract

Suppose that the finite group $G=AB$ is a mutually permutable product of two subgroups $A$ and $B$. By using Sylow numbers of $A$ and $B$, we present some new bounds of the $p$-length $l_p(G)$ of a $p$-solvable group $G$ and the nilpotent length $F_l(G)$ and the derived length $dl(G/\Phi(G))$ of a solvable group $G$. Some known results of Zhang in J. Algebra 1995, 176 are extended.