Hostname: page-component-7479d7b7d-pfhbr Total loading time: 0 Render date: 2024-07-12T16:02:43.373Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite soluble groups admitting an automorphism of prime power order with few fixed points

Published online by Cambridge University Press:  24 October 2008

Brian Hartley  and
Volker Turau
Brian Hartley
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL
Volker Turau
Affiliation:
Institut für Informatik III, Universität Karlsruhe, 7500 Karlsruhe, West Germany
Get access Rights & Permissions [Opens in a new window]

Extract

Let G be a finite soluble group with Fitting subgroup F(G). The Fitting series of G is defined, as usual, by F0(G) = 1 and Fi(G)/Fi−1(G) = F(G/Fi−1(G)) for i ≥ 1, and the Fitting height h = h(G) of G is the least integer such that Fn(G) = G. Suppose now that a finite soluble group A acts on G. Let k be the composition length of A, that is, the number of prime divisors (counting multiplicities) of |A|. There is a certain amount of evidence in favour of the

CONJECTURE. |G:Fk(G)| is bounded by a number depending only on |A| and |CG(A)|.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 3 , November 1987 , pp. 431 - 441
Copyright
Copyright © Cambridge Philosophical Society 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Gorenstein, D.. Finite Groups (Harper and Row, 1968).Google Scholar
[2]Gross, F.. Solvable groups admitting a fixed point free automorphism of prime power order. Proc. Amer. Math. Soc. 17 (1966), 1440–1446.CrossRefGoogle Scholar
[3]Gross, F.. Groups admitting a fixed-point-free automorphism of order 2n. Pacific J. Math. 24 (1968), 269–275.CrossRefGoogle Scholar
[4]Hartley, B. and Meixner, Th.. Finite soluble groups containing an element of prime order whose centralizer is small, Arch. Math. 36 (1981), 211–213.CrossRefGoogle Scholar
[5]Huppert, B.. Endliche Gruppen I (Springer-Verlag, 1967).CrossRefGoogle Scholar
[6]Meixner, Th.. Solvable groups admitting an automorphism of prime power order whose centralizer is small. J. Algebra 99 (1986), 181–190.CrossRefGoogle Scholar
[7]Robinson, D. J. S.. Finiteness conditions and generalized soluble groups, vol. 2 (Springer- Verlag, 1972).Google Scholar
[8]Roseblade, J. E.. On groups in which every subgroup is subnormal. J. Algebra 2 (1985), 402–412.CrossRefGoogle Scholar
[9]Turull, A.. Supersolvable automorphism groups of solvable groups. Math. Z. 183 (1983), 47–73.CrossRefGoogle Scholar
[10]Turull, A.. Fitting height of groups and of fixed points. J. Algebra 86 (1984), 555–566.CrossRefGoogle Scholar