The powerful class of Sylow subgroups of finite groups

TL;DR

This paper investigates how the powerful class of Sylow p-subgroups influences transfer and fusion control in finite groups, providing explicit bounds for p-length based on these subgroup properties.

Contribution

It introduces a new connection between the powerful class of Sylow p-subgroups and the control of transfer and fusion in finite groups, including explicit bounds for p-length.

Findings

Bound for p-length in terms of Sylow p-subgroup's powerful class

Explicit relationship between subgroup properties and group structure

Enhanced understanding of transfer and fusion control mechanisms

Abstract

The paper explores the effect of powerful class of Sylow $p$-subgroups of a given finite group on control of transfer or fusion. We also find an explicit bound for the $p$-length of a $p$-solvable group in terms of the poweful class of a Sylow $p$-subgroup.