Recognition of the simple groups $PSL_2(q)$ by character degree graph and order

Abstract

Let \\(G\\) be a finite group. The character degree graph of \\(G\\), which is denoted by \\(\\Gamma (G)\\), is the graph whose vertices are the prime divisors of the character degrees of the group \\(G\\) and two vertices \\(p_1\\) and \\(p_2\\) are joined by an edge if \\(p_1p_2\\) divides some character degree of \\(G\\). In this paper we prove that the simple group \\(\\mathrm{PSL}(2,p^2) \\) is uniquely determined by its character degree graph and its order. Let \\(X_1(G)\\) be the set of all irreducible complex character degrees of \\(G\\) counting multiplicities. As a consequence of our results we prove that if \\(G\\) is a finite group such that \\(X_1(G)=X_1(\\mathrm{PSL}(2,p^2) )\\), then \\(G\\cong \\mathrm{PSL}(2,p^2) \\). This implies that \\(\\mathrm{PSL}(2,p^2) \\) is uniquely determined by the structure of its complex group algebra.