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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Cyclic by prime fixed point free action


Author: Alexandre Turull
Journal: Proc. Amer. Math. Soc. 125 (1997), 3465-3470
MSC (1991): Primary 20D45
MathSciNet review: 1443859
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Abstract: Let the finite group $A$ be acting on a (solvable) group $G$ and suppose that no non-trivial element of $G$ is fixed under the action of all the elements of $A$. Assume furthermore that $(|A| , |G|) = 1$. A long standing conjecture is that then the Fitting height of $G$ is bounded by the length of the longest chain of subgroups of $A$. Even though this conjecture is known to hold for large classes of groups $A$, it is still unknown for some relatively uncomplicated groups. In the present paper we prove the conjecture for all finite groups $A$ that have a normal cyclic subgroup of square free order and prime index. Since many of these groups have natural modules where they act faithfully and coprimely but without regular orbits, the result is new for many of the groups we consider.


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Additional Information

Alexandre Turull
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: turull@math.ufl.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04263-9
PII: S 0002-9939(97)04263-9
Keywords: Solvable groups, fixed point free action, finite groups, representations
Received by editor(s): June 11, 1996
Additional Notes: Partially supported by a grant from the NSF
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1997 American Mathematical Society