Principal blocks with 5 irreducible characters

TL;DR

This paper characterizes the structure of Sylow p-subgroups in finite groups based on the number of irreducible characters in the principal p-block, revealing specific order constraints.

Contribution

It establishes a precise link between the number of irreducible characters in the principal p-block and the possible structures of Sylow p-subgroups.

Findings

Sylow p-subgroup has order 5, 7, or is a non-abelian 2-group of order 8

Principal p-block with 5 irreducible characters constrains Sylow p-subgroup structure

Provides classification criteria for Sylow p-subgroups based on block characters

Abstract

We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.