The Baby Monster is the largest group with at most $2$ irreducible characters with the same degree

TL;DR

The paper provides a complete classification of finite groups with $m(G)=2$, proving that the Baby Monster is the largest such group. It decomposes the problem into solvable, almost-simple, and unique non-abelian composition factor cases, employing Clifford theory, rationality of character fields, $p$-Brauer character analysis, and $H^2$-cohomology, supplemented by computational checks. The results yield explicit lists for the solvable and almost-simple scenarios and show that no group with multiple non-abelian composition factors can have $m(G)=2$, thereby proving the main claim and supporting Moretó's conjecture in this extremal case. The findings advance Brauer's problem by clarifying the multiplicity constraints on irreducible character degrees and identifying the Baby Monster as the maximal example under the $m(G)=2$ bound.

Abstract

We classify all finite groups such that all irreducible character degrees appear with multiplicity at most $2$. As a consequence, we prove that the largest group with at most $2$ irreducible characters of the same degree is the Baby Monster.

Theorems & Definitions (29)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4: Theorem A of Tent
• Theorem 2.5: Theorem 1.2 of Trefethen
• Lemma 2.6
• proof