November 30, 2025

Characters of relative π′ ^ ‐degree over normal subgroups

Key Points

A further extension of the Gluck–Wolf theorem was established for sets of primes, and involved normal subgroups.
Key relationships between irreducible characters of finite groups and Sylow subgroups were analyzed to prove concepts.
Using group representation theory, a broader understanding of character ratios was explored within the context of alternating groups.
The findings may lead to deeper insights into Brauer's Height Zero Conjecture, aligning with existing mathematical frameworks.

Abstract

Abstract The Gluck–Wolf theorem and its general version Navarro and Tiep, Annals of Math. 178 (2013), 1135–1171 relate arithmetic properties at a fixed prime of the ratios , for irreducible characters of a finite group that lie over a fixed ‐invariant irreducible character of a normal subgroup of , to the structure of Sylow ‐subgroups of . This result constituted a key step towards the recent proof Malle et al., Annals of Math. 200 (2024), 557–608 of Brauer's Height Zero Conjecture. In this paper, we prove a further extension of the Gluck–Wolf theorem to sets of primes, with a mild condition on if the alternating group is involved in the group.

Ask AI

Mark Helpful

Bookmark

Relay