A noncoprime Shult type theorem. (English) Zbl 0612.20008
Let G be a finite p-solvable group and let V be a faithful KG-module, K a field. Assume that G contains an element x of order \(p^ n\), p a prime, acting fixed point freely on V. Conditions are given assuring that \(x^{p^{n-1}}\in O_{p'p}(G)\). If char(K) does not divide the order of \(O_{p'}(G)\) we state results assuring that \(x^{p^{n-1}}\in O_ p(G)\). These results extend Shult’s theorem.
A new proof of Thompson’s conjecture asserting that a finite p-solvable (resp. solvable) group G admitting an automorphism \(\alpha\) of order \(p^ n\) acting fixed point freely on every \(\alpha\)-invariant p’-section of G has its p-length (resp. Fitting length) bounded by some functions of n is given for p odd. B. Hartley and A. Rae [Bull. Lond. Math. Soc. 5, 197-198 (1973; Zbl 0273.20015)] showed that \(l_ p(G)\leq 2n\) and \(f(G)\leq 2n^ 2+3n\). Here the best possible bounds \(l_ p(G)\leq n+1\) and \(f(G)\leq 2n+1\) are obtained.
A new proof of Thompson’s conjecture asserting that a finite p-solvable (resp. solvable) group G admitting an automorphism \(\alpha\) of order \(p^ n\) acting fixed point freely on every \(\alpha\)-invariant p’-section of G has its p-length (resp. Fitting length) bounded by some functions of n is given for p odd. B. Hartley and A. Rae [Bull. Lond. Math. Soc. 5, 197-198 (1973; Zbl 0273.20015)] showed that \(l_ p(G)\leq 2n\) and \(f(G)\leq 2n^ 2+3n\). Here the best possible bounds \(l_ p(G)\leq n+1\) and \(f(G)\leq 2n+1\) are obtained.
MSC:
| 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 |
| 20D45 | Automorphisms of abstract finite groups |
Keywords:
finite p-solvable group; Shult’s theorem; automorphism; acting fixed point freely; p-length; Fitting length; best possible boundsCitations:
Zbl 0273.20015References:
| [1] | Berger, T.R.: Nilpotent fixed point free automorphism groups of solvable groups. Math. Z.131, 305-312 (1973) · doi:10.1007/BF01174905 |
| [2] | Espuelas, A.: The Fitting length of the Hughes subgroup. J. Algebra105, 365-371 (1987) · Zbl 0604.20021 · doi:10.1016/0021-8693(87)90201-8 |
| [3] | Espuelas, A.: Generalized fixed point free automorphisms. Arch. Math.47, 211-214 (1986) · Zbl 0605.20015 · doi:10.1007/BF01191995 |
| [4] | Gorenstein, D.: Finite groups. New York and London: Harper and Row 1968 · Zbl 0185.05701 |
| [5] | Gross, F.: Solvable groups admitting a fixed-point-free automorphism of prime power order. Proc. Am. Math. Soc.17, 1440-1446 (1966) · Zbl 0147.27201 · doi:10.1090/S0002-9939-1966-0207836-0 |
| [6] | Hall, P., Higman, G.: On thep-length ofp-soluble groups and reduction theorems for the Burnside problem. Proc. London Math. Soc. (3)6, 1-42 (1956) · Zbl 0073.25503 · doi:10.1112/plms/s3-6.1.1 |
| [7] | Hartley, B.: Sylowp-subgroups and localp-solubility. J. Algebra23, 347-369 (1972) · Zbl 0246.20022 · doi:10.1016/0021-8693(72)90136-6 |
| [8] | Hartley, B., Rae, A.: Finitep-groups acting onp-soluble groups. Bull. London Math. Soc.5, 197-198 (1973) · Zbl 0273.20015 · doi:10.1112/blms/5.2.197 |
| [9] | Huppert, B., Blackburn, N.: Finite groups II. Berlin Heidelberg New York: Springer 1982 · Zbl 0477.20001 |
| [10] | Issacs, M.: Characters of solvable and symplectic groups. Am. J. Math.95, 594-635 (1973) · Zbl 0277.20008 · doi:10.2307/2373731 |
| [11] | Rae, A.: Sylowp-subgroups of finitep-soluble groups. J. London Math. Soc. (2)7, 117-123 (1973) · Zbl 0273.20016 · doi:10.1112/jlms/s2-7.1.117 |
| [12] | Shult, E.: On groups admitting fixed point free abelian operator groups. Illinois J. Math.9, 701-720 (1965) · Zbl 0136.28504 |
| [13] | Turull, A.: Supersolvable automorphism groups of solvable groups. Math. Z.183, 47-73 (1983) · doi:10.1007/BF01187215 |
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