Let $G$ be a finite group and $p$ a prime number. We prove that if $G$ is a finite group of order $|{\rm PSL}(2,p^2)|$ such that $G$ has an irreducible character of degree $p^2$ and we know that $G$ has no irreducible character $\theta $ such that $2p\mid \theta (1)$, then $G$ is isomorphic to ${\rm PSL}(2,p^2)$. As a consequence of our result we prove that ${\rm PSL}(2,p^2)$ is uniquely determined by the structure of its complex group algebra.
Let $G$ be a finite group. Let $X_1(G)$ be the first column of the ordinary character table of $G$. We will show that if $X_1(G)=X_1({\rm PGU}_3(q^2))$, then $G \cong{\rm PGU}_3(q^2)$. As a consequence, we show that the projective general unitary groups ${\rm PGU}_3(q^2)$ are uniquely determined by the structure of their complex group algebras., Farrokh Shirjian, Ali Iranmanesh., and Obsahuje bibliografické odkazy