Structure of finite groups with restrictions on the set of conjugacy classes sizes

TL;DR

This paper characterizes the structure of finite groups based on restrictions on their conjugacy class sizes, showing they decompose into direct products with specific properties.

Contribution

It proves that groups with conjugacy class sizes set as a product of a specific set and {1,n} are isomorphic to a direct product with particular conjugacy class size distributions.

Findings

Groups with $N(G)= ext{ extOmega} imes ext{ extbraceleft}1,n ext{ extbraceright}$ decompose into direct products.

The component $A$ has conjugacy class sizes exactly $ ext{ extOmega}$.

The component $B$ has conjugacy class sizes $ ext{ extbraceleft}1,n ext{ extbraceright}$, with $n$ a prime power.

Abstract

Let $N(G)$ be the set of conjugacy classes sizes of $G$. We prove that if $N(G)=\Omega\times \{1,n\}$ for specific set $\Omega$ of integers, then $G\simeq A\times B$ where $N(A)=\Omega$, $N(B)=\{1,n\}$, and $n$ is a power of prime.