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

  • Zohreh Sayanjali
  • Zeinab Akhlaghi
  • Behrooz KhosraviEmail author
Original Paper
  • 7 Downloads

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.

Keywords

Recognition Order of a finite group Character degree Projective special linear group 

Mathematics Subject Classification

20C15 05C25 

Notes

Acknowledgements

The authors would like to thank the referee for valuable comments and suggestions.

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Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceAmirkabir University of Technology (Tehran Polytechnic)TehranIran

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