A new characterization of L2(p2)

• Main Authors: Wang Zhongbi, Qin Chao, Lv Heng, Yan Yanxiong, Chen Guiyun
• Format: Article
• Language: English
• Published: De Gruyter 2020-09-01
• Series: Open Mathematics
• Subjects: irreducible characters; degree; maximal p-part; simple group; characterization; 20d05, 20c15
• Online Access: https://doi.org/10.1515/math-2020-0048

For a positive integer n and a prime p, let np{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 GG, ρ(G)\rho (G) the set of all prime divisors of integers in cd(G)\text{cd}(G), V(G)=pep(G)|p∈ρ(G)V(G)=\left\{{p}^{{e}_{p}(G)}|p\in \rho (G)\right\}, where pep(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≅L2(p2)G\cong {L}_{2}({p}^{2}) if and only if |G|=|L2(p2)||G|=|{L}_{2}({p}^{2})| and V(G)=V(L2(p2))V(G)=V({L}_{2}({p}^{2})).