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 be acting on a (solvable) group and suppose that no non-trivial element of is fixed under the action of all the elements of . Assume furthermore that . A long standing conjecture is that then the Fitting height of is bounded by the length of the longest chain of subgroups of . Even though this conjecture is known to hold for large classes of groups , it is still unknown for some relatively uncomplicated groups. In the present paper we prove the conjecture for all finite groups 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.

**1.**Arnold D. Feldman,*Fitting height of solvable groups admitting fixed-point-free automorphism groups*, J. Algebra**53**(1978), no. 1, 268–295. MR**0486116****2.**Alexandre Turull,*Supersolvable automorphism groups of solvable groups*, Math. Z.**183**(1983), no. 1, 47–73. MR**701358**, 10.1007/BF01187215**3.**Alexandre Turull,*Fitting height of groups and of fixed points*, J. Algebra**86**(1984), no. 2, 555–566. MR**732266**, 10.1016/0021-8693(84)90048-6**4.**Alexandre Turull,*Fixed point free action with regular orbits*, J. Reine Angew. Math.**371**(1986), 67–91. MR**859320**, 10.1515/crll.1986.371.67**5.**Alexandre Turull,*Generic fixed point free action of arbitrary finite groups*, Math. Z.**187**(1984), no. 4, 491–503. MR**760049**, 10.1007/BF01174184**6.**A. Turull,*Fixed point free action with some regular orbits*, J. Algebra (to appear).

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Additional Information

**Alexandre Turull**

Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611

Email:
turull@math.ufl.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-04263-9

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