Characters and Sylow $3$-subgroup abelianization

TL;DR

This paper characterizes finite groups with Sylow 3-subgroups having abelianization of order 9 by relating it to the number of height zero characters in the principal 3-block, settling a conjecture from 2018.

Contribution

It provides a complete characterization of groups with specific Sylow 3-subgroup properties based on character theory, confirming a conjecture by Navarro, Sambale, and Tiep.

Findings

Characterization of groups with Sylow 3-subgroup abelianization of order 9

Verification of the conjecture relating Sylow subgroup structure to block characters

Extension of Laradji's result to blocks of maximal defect without additional hypotheses

Abstract

We characterize when a finite group G possesses a Sylow 3-subgroup P with abelianization of order 9 in terms of the number of height zero characters lying in the principal 3-block of G, settling a conjecture put forward by Navarro, Sambale, and Tiep in 2018. Along the way, we show that a recent result by Laradji on the number of character of height zero in a block that lie above a given character of some normal subgroup holds, without any hypothesis on the group for blocks of maximal defect.