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Groups of automorphisms and centralizers

Published online by Cambridge University Press:  24 October 2008

Alexandre Turull
Alexandre Turull
Affiliation:
Department of Mathematics and Computer Science, University of Miami, Coral Gables, Florida 33124, U.S.A.
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Let G be a finite solvable group and A a group of automorphisms of G such that (|A|, |G|) = 1. We denote by h(G) the Fitting height of G and by l(A) the length of the longest chain of subgroups of A. Then, under some additional hypotheses, it is known from [5] that h(G) ≤ 2l(A) + h(CG(A)) and from [8] that, when CG(A) = 1, h(G) ≤ l(A), both results being best possible (see [6, 7]). The present paper attempts to explain the difference in the coefficient of l(A) in the two inequalities, from 2 to 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 2 , March 1990 , pp. 227 - 238
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Hartley, B. and Isaacs, I. M.. On characters and fixed points of coprime operator groups. (To appear.)Google Scholar
[2]Hartley, B. and Turau, V.. Finite soluble groups admitting an automorphism of prime power order with few fixed points. Math. Proc. Cambridge Philos. Soc. 102 (1987), 431–441CrossRefGoogle Scholar
[3]Isaacs, I. M.. Character Theory of Finite Groups (Academic Press, 1976).Google Scholar
[4]Meixner, T.. Solvable groups admitting an automorphism of prime power order whose centralizer is small. J. Algebra 99 (1986), 181–190.CrossRefGoogle Scholar
[5]Turull, A.. Fitting height of groups and of fixed points. J. Algebra 86 (1984), 555–566.CrossRefGoogle Scholar
[6]Turull, A.. Examples of centralizers of automorphism groups. Proc. Amer. Math. Soc. 91 (1984), 537–539.CrossRefGoogle Scholar
[7]Turull, A., Generic fixed point free action of arbitrary finite groups. Math. Z. 187 (1984), 491–503.CrossRefGoogle Scholar
[8]Turull, A., Fixed point free action with regular orbits. J. Reine Angew. Math. 371 (1986), 67–91.Google Scholar