Characters, Commutators and Centers of Sylow Subgroups

TL;DR

This paper investigates how the character table of a finite group reveals properties of its Sylow p-subgroups, providing a classification based on the generalized Fitting subgroup.

Contribution

It offers a detailed classification of groups with specific Sylow subgroup properties using character theory and the generalized Fitting subgroup.

Findings

Character table determines subgroup index properties

Classification of groups with certain Sylow subgroup structures

Insights into the relationship between characters and group structure

Abstract

The character table of a finite group G determines whether |P:P'|=p^2 and whether |P:Z(P)|=p^2, where P is a Sylow p-subgroup of G. To prove the latter, we give a detailed classification of those groups in terms of the generalized Fitting subgroup.