# Classification of finite C{\theta}{\theta}-groups with even order and its application

## TL;DR

This paper classifies finite Cθθ-groups with even order, explores their degree patterns, and demonstrates that certain groups like PSL(2, q) and PGL(2, q) are uniquely identified by these patterns.

## Contribution

It provides a classification of finite Cθθ-groups with even order and shows the uniqueness of specific groups based on their degree patterns.

## Key findings

- Classified finite Cθθ-groups with even order.
- Proved infinitely many Cθθ-groups share the same degree pattern.
- Demonstrated that PSL(2, q) and PGL(2, q) are determined by their degree pattern.

## Abstract

A finite group of order divisible by 3 in which centralizers of 3-elements are 3-subgroups will be called a C{\theta}{\theta}-group. The prime graph (or Gruenberg-Kegel graph) of a finite group G is denoted by {\Gamma}(G) (or GK(G)) and its a familiar. Also the degrees sequence of {\Gamma}(G) is called the degree pattern of G and is denoted by D(G). In this paper, first we classify the finite C{\theta}{\theta}-groups with even order. Then we show that there are infinitely many C{\theta}{\theta}-groups with the same degree pattern. Finally, we proved that the simple group PSL(2, q) and the almost simple group PGL(2, q), where q > 9 is a power of 3, are determined by their degree pattern.