Recognition of ${\mathrm {PSL}}(2, p^{2})$ 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.