Nonvanishing elements for Brauer characters. (English) Zbl 1377.20007
Let \(p\) be a prime such that \(p>3\), let \(G\) be a finite solvable group with \(O_p(G)=1\), and let \(g\) be a \(p\)-regular element in \(G\) such that \(\varphi(g) \neq 0\) for every irreducible \(p\)-Brauer character \(\varphi\) of \(G\). The authors prove that \(g\in O_{p'pp'}(G)\), unless \(p\in \{5,7\}\), and the order of \(g\) is divisible by 2 or 3.
Reviewer: Burkhard Külshammer (Jena)
MSC:
| 20C20 | Modular representations and characters |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
References:
| [1] | S.Dolfi, ‘Orbits of permutation groups on the power set’, Arch. Math.75 (2000), 321-327.10.1007/s0001300505101785438 · Zbl 0978.20001 · doi:10.1007/s000130050510 |
| [2] | S.Dolfi, ‘Large orbits in coprime actions of solvable groups’, Trans. Amer. Math. Soc.360 (2008), 135-152.10.1090/S0002-9947-07-04155-42341997 · Zbl 1129.20016 · doi:10.1090/S0002-9947-07-04155-4 |
| [3] | S.Dolfi, G.Navarro, E.Pacifici, L.Sanus and P. H.Tiep, ‘Non-vanishing elements of finite groups’, J. Algebra323 (2010), 540-545.10.1016/j.jalgebra.2009.08.0142564856 · Zbl 1192.20006 · doi:10.1016/j.jalgebra.2009.08.014 |
| [4] | S.Dolfi and E.Pacifici, ‘Zeros of Brauer characters and linear actions of finite groups’, J. Algebra340 (2011), 104-113.10.1016/j.jalgebra.2011.05.0232813565 · Zbl 1253.20007 · doi:10.1016/j.jalgebra.2011.05.023 |
| [5] | D.Foulser, ‘Solvable primitive permutation groups of low rank’, Trans. Amer. Math. Soc.143 (1969), 1-54.10.1090/S0002-9947-1969-0257199-70257199 · Zbl 0187.29401 · doi:10.1090/S0002-9947-1969-0257199-7 |
| [6] | I. M.Isaacs, G.Navarro and T. R.Wolf, ‘Finite group elements where no irreducible character vanishes’, J. Algebra222 (1999), 413-423.10.1006/jabr.1999.80071733678 · Zbl 0959.20009 · doi:10.1006/jabr.1999.8007 |
| [7] | O.Manz and T. R.Wolf, Representations of Solvable Groups (Cambridge University Press, Cambridge, 1993).10.1017/CBO97805115259711261638 · Zbl 0928.20008 · doi:10.1017/CBO9780511525971 |
| [8] | G.Navarro, Characters and Blocks of Finite Groups (Cambridge University Press, Cambridge, 1998).10.1017/CBO97805115260151632299 · Zbl 0903.20004 · doi:10.1017/CBO9780511526015 |
| [9] | A.Seress, ‘The minimal base size of primitive solvable permutation groups’, J. Lond. Math. Soc. (2)53 (1996), 243-255.10.1112/jlms/53.2.2431373058 · Zbl 0854.20004 · doi:10.1112/jlms/53.2.243 |
| [10] | Y.Yang, ‘Regular orbits of finite primitive solvable groups, II’, J. Algebra341 (2011), 23-34.10.1016/j.jalgebra.2011.03.0312824509 · Zbl 1247.20021 · doi:10.1016/j.jalgebra.2011.03.031 |
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.
