Finite groups in which almost all class sizes are prime numbers

TL;DR

This work analyzes finite groups whose conjugacy-class sizes contain exactly two composite numbers. By exploiting coprime-class-size phenomena and centraliser structure, the authors prove that such groups have at most three prime class sizes and obtain a precise structural decomposition up to abelian direct factors into a $p$-group piece of nilpotency class at most 2 and a Frobenius-type component. They establish solubility in key cases and classify possible class-size sets when the total number of class sizes is 4 or 5, including detailed configurations and the role of the Fitting subgroup $F_2(G)$. The results also exclude almost-simple groups in this setting and provide concrete examples illustrating realizable patterns, connecting the problem to analogous questions for irreducible character degrees.

Abstract

In this paper we study arithmetical and structural features of a finite group that possesses exactly two conjugacy class sizes that are composite numbers.

Theorems & Definitions (13)

• Theorem 1
• Conjecture 2
• Theorem 3
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Lemma 2.4
• Theorem 2.5
• Lemma 2.6
• Proposition 3.1