Recognition of \(\text{PSL}(2,p)\) by order and some information on its character degrees where \(p\) is a prime.

*(English)*Zbl 1304.20042Summary: Let \(G\) be a finite group and \(\text{cd}(G)\) be the set of irreducible character degrees of \(G\). In this paper we prove that if \(p\) is a prime number, then the simple group \(\text{PSL}(2,p)\) is uniquely determined by its order and some information about its character degrees. In fact we prove that if \(G\) is a finite group such that (i) \(|G|=|\text{PSL}(2,p)|\), (ii) \(p\in\text{cd}(G)\), (iii) \(\text{cd}(G)\) has an even integer, and (iv) there does not exist any element \(a\in\text{cd}(G)\) such that \(2p\mid a\), then \(G\cong\text{PSL}(2,p)\). As a consequence of our result we get that \(\text{PSL}(2,p)\) is uniquely determined by its order and the largest and the second largest character degrees.

##### MSC:

20D60 | Arithmetic and combinatorial problems involving abstract finite groups |

20C15 | Ordinary representations and characters |

20D06 | Simple groups: alternating groups and groups of Lie type |

20C33 | Representations of finite groups of Lie type |

##### Keywords:

finite groups; sets of irreducible character degrees; large character degrees; orders; projective special linear groups; recognition
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\textit{B. Khosravi} et al., Monatsh. Math. 175, No. 2, 277--282 (2014; Zbl 1304.20042)

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##### References:

[1] | Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, Oxford (1985) · Zbl 0568.20001 |

[2] | Huppert, B.: Character Theory of Finite Groups. de Gruyter, Berlin (1998) · Zbl 0932.20007 |

[3] | Huppert, B, Some simple groups which are determined by the set of their character degrees (I), Ill. J. Math., 44, 828-842, (2000) · Zbl 0972.20006 |

[4] | Isaacs, I.M.: Character Theory of Finite Groups. Academic Press, New York (1976) · Zbl 0337.20005 |

[5] | Isaacs, IM, Character degree graphs and normal subgroups, Trans. Am. Math. Soc., 356, 1155-1183, (2004) · Zbl 1034.20009 |

[6] | Khosravi, B.: Groups with the same order and large character degrees as \({\rm PGL(2,9)}\), Quasigr. Relat. Syst. (to appear) |

[7] | Lewis, ML; White, DL, Nonsolvable groups with no prime dividing three character degrees, J. Algebra, 336, 158-183, (2011) · Zbl 1246.20006 |

[8] | Xu, H., Chen, G.Y., Yan, Y.: A new characterization of simple \(K_3\)-groups by their orders and large degrees of their irreducible characters. Comm. Algebra (to appear) · Zbl 1297.20012 |

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