##
**On distinct character degrees.**
*(English)*
Zbl 1131.20006

Let \(G\) be a finite group and let \(N\) be a normal subgroup of \(G\). The general problem is to determine those pairs such that the irreducible characters of \(G\) which do not have \(N\) in the kernel have distinct degrees, the author says that the pair has property (D). This was inspired by the work of Y. Berkovich, D. Chillag and M. Herzog [Proc. Am. Math. Soc. 115, No. 4, 955-959 (1992; Zbl 0822.20004)]; note the reference in the paper is wrong. They consider the situation when all the non-linear irreducible characters have distinct degrees, which is equivalent to choosing \(N\) to be the derived group. Their work derives from that of G. Seitz [Proc. Am. Math. Soc. 19, 459-461 (1968; Zbl 0244.20010)].

In this paper, the author considers what happens when \(N\) is a minimal normal subgroup of \(G\). This has already been considered in the special case that \(N\) was a unique minimal normal subgroup by Y. Berkovich, I. M. Isaacs and L. Kazarin [J. Algebra 216, No. 2, 448-480 (1999; Zbl 0939.20005)].

The main theorem of the paper proves that if \(G\) is a soluble group and \(N\) is a minimal normal subgroup of \(G\) with \(|N|=p^a\) for some prime \(p\) and \((G,N)\) has property (D) then \((G,N)\) is a Camina pair and \(O_{p'}(G)=1\). Rather more precise information can be given but they are too detailed to give in this review. Notice that if \(M>N\) is a normal subgroup and \((G,N)\) has property (D) so has \((G,M)\). So some of these results apply to any normal \(M\).

The author also discusses a conjecture due to R. J. Higgs [Glasg. Math. J. 40, No. 3, 431-434 (1998; Zbl 0927.20007)], which was originally framed as a question about projective characters.

In this paper, the author considers what happens when \(N\) is a minimal normal subgroup of \(G\). This has already been considered in the special case that \(N\) was a unique minimal normal subgroup by Y. Berkovich, I. M. Isaacs and L. Kazarin [J. Algebra 216, No. 2, 448-480 (1999; Zbl 0939.20005)].

The main theorem of the paper proves that if \(G\) is a soluble group and \(N\) is a minimal normal subgroup of \(G\) with \(|N|=p^a\) for some prime \(p\) and \((G,N)\) has property (D) then \((G,N)\) is a Camina pair and \(O_{p'}(G)=1\). Rather more precise information can be given but they are too detailed to give in this review. Notice that if \(M>N\) is a normal subgroup and \((G,N)\) has property (D) so has \((G,M)\). So some of these results apply to any normal \(M\).

The author also discusses a conjecture due to R. J. Higgs [Glasg. Math. J. 40, No. 3, 431-434 (1998; Zbl 0927.20007)], which was originally framed as a question about projective characters.

Reviewer: Alan R. Camina (Norwich)

### MSC:

20C15 | Ordinary representations and characters |

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |

### Keywords:

irreducible characters; finite groups; character degrees; Camina pairs; finite soluble groups; minimal normal subgroups### References:

[1] | Y. Berkovich, D. Chillag and M. Herzog, Finite groups in which all nonlinear irreducible characters have distinct degrees, Proceedings of the American Mathematical Society 106 (1992), 3263–3268. · Zbl 0822.20004 |

[2] | Y. Berkovich, I. M. Isaacs and L. Kazarin, Groups with distinct monolithic character degrees, Journal of Algebra 216 (1999), 448–480. · Zbl 0939.20005 |

[3] | A. R. Camina, Some conditions which almost characterize Frobenius groups, Israel Journal of Mathematics 31 (1978), 153–160. · Zbl 0654.20019 |

[4] | D. Chillag and I. D. Macdonald, Generalized Frobenius groups, Israel Journal of Mathematics 47 (1984), 111–122. · Zbl 0533.20012 |

[5] | A. Espuelas, Large character degrees of groups of odd order, Illinois Journal of Mathematics 35 (1991), 499–505. · Zbl 0713.20007 |

[6] | R. J. Higgs, On projective characters of the same degree, Glasgow Mathematical Journal 40 (1998), 431–434. · Zbl 0927.20007 |

[7] | I. M. Isaacs, Characters of Finite Groups, Dover, New York, 1994. · Zbl 0873.20005 |

[8] | I. M. Isaacs, Blocks with just two irreducible Brauer characters in solvable groups, Journal of Algebra 170 (1994), 487–503. · Zbl 0838.20012 |

[9] | E. B. Kuisch, Sylow p-subgroups of solvable Camina pairs, Journal of Algebra 156 (1993), 395–406. · Zbl 0782.20010 |

[10] | O. Manz and T. Wolf, Representations of Solvable Groups, London Mathematical Society Lecture Note Series 185, Cambridge University Press, 1993. · Zbl 0928.20008 |

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