##
**Zeros of real irreducible characters of finite groups.**
*(English)*
Zbl 1295.20006

It is proved in this paper that if all real-valued irreducible characters of a finite group \(G\) with Frobenius-Schur indicator 1 are nonzero at all 2-elements of \(G\), then \(G\) has a normal Sylow 2-subgroup. This result extends the Itô-Michler theorem (for the prime 2 and real, absolutely irreducible representations), as well as several recent results on nonvanishing elements of finite groups, such as work by S. Dolfi, G. Navarro and P. H. Tiep [Math. Z. 259, No. 4, 755-774 (2008; Zbl 1149.20006)], several papers by S. Dolfi in the last five years, and the paper by S. Dolfi, E. Pacifici, L. Sanus and P. Spiga [see J. Algebra 321, No. 1, 345-352 (2009; Zbl 1162.20005)]. As a matter of fact, the paper contains also forty references (highly recommended!).

To the convenience of the reader, we remind that the Frobenius-Schur indicator of an irreducible character \(\chi\) of a group \(G\) is precisely equal to 1, when \(\chi\) is afforded by a real representation of \(G\).

The main result of this paper shows, that, if \(\chi(x)\neq 0\) for all real-valued irreducible characters of the finite group \(G\) with Frobenius-Schur indicator 1 and all 2-elements \(x\in G\), \(G\) admits a normal Sylow 2-group. As such, this result is a generalization of [S. Dolfi, G. Navarro and P. H. Tiep (loc. cit.), Theorem A] and of other theorems. An immediate consequence is the result, that, if \(\chi(1)\) is odd for all real-valued irreducible characters \(\chi\) of \(G\) with Frobenius-Schur indicator 1, \(G\) admits a normal Sylow 2-group.

A substantial part of the paper deals with the finite simple groups. In order to obtain the main results mentioned above, the authors need the following theorem (stated in full just below) whose proof occupies about nineteen pages of the paper, thereby carefully analysing specific characters of finite simple groups:

Theorem 3.1. (paraphrased) For any \(H\) with \(S\leq H\leq\operatorname{Aut}(S)\) (\(S\) any finite non-Abelian simple group), there exists \(\chi\in\text{Irr}(S)\) and a 2-element \(x\in S\) satisfying both the following two situations:

(1) \(\chi(\alpha(x))=0\) for all \(\alpha\in\operatorname{Aut}(S)\);

(2) There exists a subgroup \(J\) with \(I_H(\chi)\leq J\leq H\) (\(I_H(\chi):=\) the inertia group of \(\chi\) in \(H\)) and an irreducible strongly real character of \(J\) lying over \(\chi\).

In summary: an important paper, but notice that the title is too modest and does not give justice to the bunch of obtained results here.

To the convenience of the reader, we remind that the Frobenius-Schur indicator of an irreducible character \(\chi\) of a group \(G\) is precisely equal to 1, when \(\chi\) is afforded by a real representation of \(G\).

The main result of this paper shows, that, if \(\chi(x)\neq 0\) for all real-valued irreducible characters of the finite group \(G\) with Frobenius-Schur indicator 1 and all 2-elements \(x\in G\), \(G\) admits a normal Sylow 2-group. As such, this result is a generalization of [S. Dolfi, G. Navarro and P. H. Tiep (loc. cit.), Theorem A] and of other theorems. An immediate consequence is the result, that, if \(\chi(1)\) is odd for all real-valued irreducible characters \(\chi\) of \(G\) with Frobenius-Schur indicator 1, \(G\) admits a normal Sylow 2-group.

A substantial part of the paper deals with the finite simple groups. In order to obtain the main results mentioned above, the authors need the following theorem (stated in full just below) whose proof occupies about nineteen pages of the paper, thereby carefully analysing specific characters of finite simple groups:

Theorem 3.1. (paraphrased) For any \(H\) with \(S\leq H\leq\operatorname{Aut}(S)\) (\(S\) any finite non-Abelian simple group), there exists \(\chi\in\text{Irr}(S)\) and a 2-element \(x\in S\) satisfying both the following two situations:

(1) \(\chi(\alpha(x))=0\) for all \(\alpha\in\operatorname{Aut}(S)\);

(2) There exists a subgroup \(J\) with \(I_H(\chi)\leq J\leq H\) (\(I_H(\chi):=\) the inertia group of \(\chi\) in \(H\)) and an irreducible strongly real character of \(J\) lying over \(\chi\).

In summary: an important paper, but notice that the title is too modest and does not give justice to the bunch of obtained results here.

Reviewer: Robert W. van der Waall (Huizen)

### MSC:

20C15 | Ordinary representations and characters |

20C33 | Representations of finite groups of Lie type |

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