A new characterization of L2(p3)

Abstract

For a positive integer n and a prime p, let n_p denote the p-part of n. Let G be a group, $\text{cd}(G)$ the set of all irreducible character degrees of G, $\rho (G)$ the set of all prime divisors of integers in $\text{cd}(G),V(G)=\{p^{e_p(G)}\mid p\in \rho (G)\},$ where $p^{e_p(G)}=\text{max}\{\chi {(1)}_p\mid \chi \in \text{Irr}(\mathrm{G})\}.$ The authors proved that $G\cong L_2(p^2)$ if and only if $\left|G\right|=|L_2(p^2)|$ and $V(G)=V(L_2(p^2)).$ In this paper, the authors continue this topic and prove that $G\cong L_2(p^3)$ if and only if $\left|G\right|=|L_2(p^3)|$ and $V(G)=V(L_2(p^3)).$

Keywords:

• Characterization
• degree
• irreducible characters
• maximal p-part
• simple group

2020 Mathematics Subject Classification:

• 20D05
• 20C15

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [Grant Nos. 12071376, 11971391], Fundamental Research Funds for the Central Universities [No. XDJK2019B030].