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The existence of regular orbits.(English)Zbl 0654.20003

Let G be a solvable group and p an odd prime. It is shown that if A is a p-subgroup of G and X is a p-chief factor of G on which A acts faithfully, then X contains a regular A-orbit, i.e., there exists \(x\in X\) such that \(C_ A(x)=1\). The result is false for \(p=2\). As a corollary, the following character-theoretic result is obtained: Let G be a solvable primitive group. Assume that F(G) is a p-group, p odd. Take \(P\in S_ p(G).Then\) \(b(P)=\max \{\psi (1)|\) \(\psi \in Irr(P)\}=| G:F(G)|_ p\).
Reviewer: A.Espuelas

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure

Cite

References:

[1]Berger, T. R.: Hall-higman type theorems, V. Pacific J. Math. 73, 1-62 (1977) ·Zbl 0412.20004
[2]Berger, T. R.: Hall-higman type theorems, VI. J. algebra 51, 416-424 (1978) ·Zbl 0377.20007
[3]Berger, T. R.: On the structure of a representation of a finite solvable group. Vorlesungen fachbereich math. Univ. essen 12 (1985) ·Zbl 0562.20002
[4]Fleischmann, P.: Finite groups with regular orbits on vector spaces. J. algebra 103, 211-215 (1986) ·Zbl 0604.20014
[5]Gow, R.: On the number of characters in a p-block of a p-solvable group. J. algebra 65, 421-426 (1980) ·Zbl 0445.20003
[6]Hall, P.; Higman, G.: On the p-length of p-soluble groups and reduction theorems for the Burnside problem. Proc. London math. Soc. (3) 6, 1-42 (1956) ·Zbl 0073.25503
[7]Hargraves, B. B.: The existence of regular orbits for nilpotent groups. J. algebra 72, 54-101 (1981) ·Zbl 0482.20004
[8]Huppert, B.; Blackburn, N.: Finite groups, II. (1982) ·Zbl 0477.20001
[9]Jones, G.: The influence of nilpotent subgroups on the nilpotent length and the derived length of a finite group. Proc. London math. Soc. (3) 49, 343-360 (1984) ·Zbl 0542.20006
[10]Turull, A.: Supersolvable automorphism groups of solvable groups. Math. Z. 183, 47-73 (1983) ·Zbl 0496.20016
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.
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