A new characterization of L 2(p 2)

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})) .