A new characterization of L 2(p 2)

Zhongbi Wang,Chao Qin,Guiyun Chen,Yanxiong Yan,Heng Lv

doi:10.1515/math-2020-0048

Abstract

Abstract For a positive integer n and a prime p, let n p {n}_{p} denote the p-part of n. Let G be a group, cd ( G ) \text{cd}(G) the set of all irreducible character degrees of G G , ρ ( G ) \rho (G) the set of all prime divisors of integers in cd ( G ) \text{cd}(G) , V ( G ) = p e p ( G ) | p ∈ ρ ( G ) V(G)=\left\{{p}^{{e}_{p}(G)}|p\in \rho (G)\right\} , where p e p ( G ) = max { χ ( 1 ) p | χ ∈ Irr ( G ) } . {p}^{{e}_{p}(G)}=\hspace{.25em}\max \hspace{.25em}\{\chi {(1)}_{p}|\chi \in \text{Irr}(G)\}. In this article, it is proved that G ≅ L 2 ( p 2 ) G\cong {L}_{2}({p}^{2}) if and only if | G | = | L 2 ( p 2 ) | |G|=|{L}_{2}({p}^{2})| and V ( G ) = V ( L 2 ( p 2 ) ) V(G)=V({L}_{2}({p}^{2})) .

Highlights

In what follows, the notations are the same as in [1]
An interesting fact is that if Huppert’s conjecture has been proved, it is natural that all finite non-abelian simple groups can be uniquely determined by their orders and the sets of irreducible character degrees
It has been proved that many simple groups were determined by their orders and character degree graph

Summary

Introduction

The notations are the same as in [1]. Let G be a finite group, |G| denotes the order of G, Irr(G) set of all irreducible characters of G, and cd(G) denotes the set of irreducible character degrees of G. An interesting fact is that if Huppert’s conjecture has been proved, it is natural that all finite non-abelian simple groups can be uniquely determined by their orders and the sets of irreducible character degrees. It has been proved that many simple groups were determined by their orders and character degree graph.

Results

Conclusion