Characterization of $${\textit{PGL}}(2 , p^{2} ) $$ by Order and Some Irreducible Character Degrees

Abstract

In this paper, we determine all of finite groups whose order and the largest of their irreducible character degrees are the same as $${\textit{PGL}}( 2 , p^{2} ) $$ for all odd prime numbers p. As a consequence, we show that the groups $${\textit{PGL}}( 2 , p^{2} ) $$ are uniquely determined by the structure of their complex group algebras.