Recognition of $${\mathrm {PSL}}(2, p^{2})$$ PSL ( 2 , p 2 ) by Order and Some Properties on Character Degrees

Abstract

Let G be a finite group and $${\mathrm {Irr}}(G)$$ be the set of all irreducible complex characters of G. Furthermore $${\mathrm {cd}}(G)$$ is the set of all character degrees of G. In this paper, we introduce a new characterization of $${\mathrm {PSL}}(2, p^{2})$$ , where p is an odd prime number, by order and some properties on character degrees. In fact, we prove that if $$|G |=| {\mathrm {PSL}}(2, p^{2})| $$ and $$p^{2}+1 \in {\mathrm {cd}}(G)$$ and there exists no $$\theta \in {\mathrm {Irr}}(G)$$ such that $$2p \mid \theta (1)$$ , then $$G \cong {\mathrm {PSL}}(2, p^{2}) $$ . Also by an example we show that $$ {\mathrm {PSL}}(2, p^{2}) $$ , where p is an odd prime, is not recognizable by order and the largest character degree.