Groups with 2-generated Sylow subgroups and their character tables

TL;DR

This paper investigates how the character table of a finite group reveals properties of its Sylow p-subgroups, including their nilpotency class, minimal non-abelian structure, and two-generation status.

Contribution

It establishes that character tables can determine key structural features of Sylow p-subgroups, providing a classification framework for these groups.

Findings

Character table determines if P has maximal nilpotency class

Character table reveals if P is minimal non-abelian

For p-constrained groups, character table indicates if P is 2-generated

Abstract

Let G be a finite group with Sylow p-subgroup P. We show that the character table of G determines whether P has maximal nilpotency class and whether P is a minimal non-abelian group. The latter result is obtained from a precise classification of the corresponding groups G in terms of their composition factors. For p-constrained groups G we prove further that the character table determines whether P can be generated by two elements.