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

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## 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.

## Keywords

Almost simple groups Character degree Order Projective general linear group Complex group algebras## Mathematics Subject Classification

20C15 20D05 20D60## Notes

### Acknowledgements

The authors express their gratitude to the referees for very useful comments and suggestions. Partial support by the Center of Excellence of Algebraic Structures and its Applications of Tarbiat Modares University (CEAS) is gratefully acknowledged by the first author (AI)

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© Iranian Mathematical Society 2019