Irreducible characters which are zero on only one conjugacy class.

*(English)*Zbl 1112.20007Let \(G\) be a finite group and let \(\chi\) be an irreducible complex character of \(G\) of degree greater than 1. A theorem of Burnside shows that \(\chi\) vanishes on at least one conjugacy class of \(G\). As its title suggests, this paper is concerned with investigating what can be said when \(\chi\) vanishes on exactly one conjugacy class. The available examples suggest that the phenomenon is related to certain doubly transitive groups. For example, when \(q\) is a power of a prime \(p\), the Steinberg character of the group \(\text{PGL}(2,q)\) is an irreducible character of degree \(q\), derived from a doubly transitive action on \(q+1\) points, which vanishes only on elements of order \(p\). Since all elements of order \(p\) are conjugate in \(\text{PGL}(2,q)\), we have a group with the required property, and this group is simple when \(q\) is a power of 2. There are one or two other simple groups with the unique conjugacy class property, but we do not know of any infinite families except that described above.

The authors’ main theorem is that if \(G\) is a finite solvable group with a character of the type described above, it has a homomorphic image which is a doubly transitive group. In this case, \(\chi(1)+1\) is a power of a prime. This result is deduced from a more general hypothesis concerning the existence of an Abelian chief factor in \(G\), which automatically holds if \(G\) is solvable. It seems that more might be said, as the role of the kernel of \(\chi\) is not fully exploited. For if \(\chi(g)=0\) and \(h\) is any element of \(\ker\chi\), then \(\chi(gh)=0\) also, and hence \(g\) and \(gh\) are conjugate in \(G\). This is a condition which may tell us something about the structure of \(\ker\chi\) or the action of \(G\) on this normal subgroup. For example, if \(g\) is an involution, the kernel must be Abelian.

The authors’ main theorem is that if \(G\) is a finite solvable group with a character of the type described above, it has a homomorphic image which is a doubly transitive group. In this case, \(\chi(1)+1\) is a power of a prime. This result is deduced from a more general hypothesis concerning the existence of an Abelian chief factor in \(G\), which automatically holds if \(G\) is solvable. It seems that more might be said, as the role of the kernel of \(\chi\) is not fully exploited. For if \(\chi(g)=0\) and \(h\) is any element of \(\ker\chi\), then \(\chi(gh)=0\) also, and hence \(g\) and \(gh\) are conjugate in \(G\). This is a condition which may tell us something about the structure of \(\ker\chi\) or the action of \(G\) on this normal subgroup. For example, if \(g\) is an involution, the kernel must be Abelian.

Reviewer: Roderick Gow (Dublin)

##### MSC:

20C15 | Ordinary representations and characters |

##### Software:

GAP
PDF
BibTeX
XML
Cite

\textit{J. D. Dixon} and \textit{A. Rahnamai Barghi}, Proc. Am. Math. Soc. 135, No. 1, 41--45 (2007; Zbl 1112.20007)

Full Text:
DOI

##### References:

[1] | Ya. G. Berkovich and E. M. Zhmud\(^{\prime}\), Characters of finite groups. Part 2, Translations of Mathematical Monographs, vol. 181, American Mathematical Society, Providence, RI, 1999. Translated from the Russian manuscript by P. Shumyatsky [P. V. Shumyatskiĭ], V. Zobina and Berkovich. |

[2] | David Chillag, On zeroes of characters of finite groups, Proc. Amer. Math. Soc. 127 (1999), no. 4, 977 – 983. · Zbl 0917.20007 |

[3] | J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. · Zbl 0568.20001 |

[4] | The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.4.4 (2005) (http://www.gap-system.org). |

[5] | Christoph Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order, Geometriae Dedicata 2 (1974), 425 – 460. · Zbl 0292.20045 |

[6] | Christoph Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II, J. Algebra 93 (1985), no. 1, 151 – 164. · Zbl 0583.20003 |

[7] | B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). · Zbl 0217.07201 |

[8] | Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. Bertram Huppert and Norman Blackburn, Finite groups. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 243, Springer-Verlag, Berlin-New York, 1982. |

[9] | I. Martin Isaacs, Character theory of finite groups, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. · Zbl 0337.20005 |

[10] | I. M. Isaacs, Character degrees and derived length of a solvable group, Canad. J. Math. 27 (1975), 146 – 151. · Zbl 0306.20008 |

[11] | Èá\textonesuperior EUR. Žmud\(^{\prime}\), Finite groups having an irreducible complex character with a class of zeros, Dokl. Akad. Nauk SSSR 247 (1979), no. 4, 788 – 791 (Russian). |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.