Two dual questions on zeros of characters of finite groups. (English) Zbl 1159.20010
The main result of the paper is the following: Let \(G\) be a finite nonabelian group. For any irreducible character \(\chi\) of \(G\), let \(v(\chi)=\{g\in G\mid\chi(g)=0\}\). Then the following three statements are equivalent:
(i) \(v(\chi)\) is a conjugacy class of \(G\) for all but one of the nonlinear irreducible characters \(\chi\) of \(G\);
(ii) all but one of the columns of the character table of \(G\) have at most one zero entry;
(iii) \(G\) satisfies one of the following: (1) \(G\) is a 2-transitive Frobenius group with kernel \(G'\), (2) \(G\) is an extraspecial 2-group, (3) \(G\) is a Frobenius group with kernel \(G'\) of order greater than 3 and complement of order 2, (4) \(G\cong\text{SL}(2,3)\), (5) \(G\cong S_4\), or (6) \(G\cong A_5\).
This solves a problem posed by Y. Berkovich and L. Kazarin in 1998. The proof depends on CFSG.
(i) \(v(\chi)\) is a conjugacy class of \(G\) for all but one of the nonlinear irreducible characters \(\chi\) of \(G\);
(ii) all but one of the columns of the character table of \(G\) have at most one zero entry;
(iii) \(G\) satisfies one of the following: (1) \(G\) is a 2-transitive Frobenius group with kernel \(G'\), (2) \(G\) is an extraspecial 2-group, (3) \(G\) is a Frobenius group with kernel \(G'\) of order greater than 3 and complement of order 2, (4) \(G\cong\text{SL}(2,3)\), (5) \(G\cong S_4\), or (6) \(G\cong A_5\).
This solves a problem posed by Y. Berkovich and L. Kazarin in 1998. The proof depends on CFSG.
Reviewer: Thomas Michael Keller (San Marcos)
MSC:
| 20C15 | Ordinary representations and characters |
Keywords:
finite groups; vanishing subsets; character tables; nonlinear characters; irreducible charactersReferences:
| [1] | Berkovich Y., Houston J. Math. 24 pp 619– (1998) |
| [2] | Bianchi M., Houston J. Math. 26 pp 451– (2000) |
| [3] | DOI: 10.1090/S0002-9939-99-04790-5 · Zbl 0917.20007 · doi:10.1090/S0002-9939-99-04790-5 |
| [4] | Feit W., Nagoya Math. J. 21 pp 185– (1962) |
| [5] | Isaacs I. M., Canad. J. Math. 41 pp 68– (1989) · Zbl 0686.20002 · doi:10.4153/CJM-1989-003-2 |
| [6] | DOI: 10.1515/jgth.2000.028 · Zbl 0965.20003 · doi:10.1515/jgth.2000.028 |
| [7] | DOI: 10.1023/A:1022676707499 · doi:10.1023/A:1022676707499 |
| [8] | DOI: 10.1006/jabr.1995.1213 · Zbl 0839.20014 · doi:10.1006/jabr.1995.1213 |
| [9] | DOI: 10.2307/2035551 · Zbl 0244.20010 · doi:10.2307/2035551 |
| [10] | DOI: 10.2307/1993556 · Zbl 0101.01604 · doi:10.2307/1993556 |
| [11] | DOI: 10.1007/BF02764723 · Zbl 0582.20014 · doi:10.1007/BF02764723 |
| [12] | DOI: 10.1016/0021-8693(88)90176-7 · Zbl 0653.20014 · doi:10.1016/0021-8693(88)90176-7 |
| [13] | DOI: 10.1016/0021-8693(81)90218-0 · Zbl 0471.20013 · doi:10.1016/0021-8693(81)90218-0 |
| [14] | DOI: 10.1017/S1446788700004511 · Zbl 0203.02902 · doi:10.1017/S1446788700004511 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
