Frobenius groups of automorphisms and their fixed points
-
Evgeny Khukhro
,
Natalia Makarenko
Abstract.
Suppose that a finite group G admits a Frobenius group of
automorphisms 
with kernel F and complement H such that
the fixed-point subgroup of F is trivial: 
. In this
situation various properties of G are shown to be close to the corresponding
properties of 
. By using Clifford's theorem
it is proved that the order 
is bounded in terms
of 
and 
, the rank of G is bounded in terms
of and the rank of , and
that G is nilpotent if is nilpotent. Lie ring methods
are used for bounding the exponent
and the nilpotency class
of G in the case of metacyclic . The exponent of G is
bounded in terms of 
and the exponent of by using Lazard's
Lie algebra associated with the Jennings–Zassenhaus
filtration and its connection with powerful subgroups. The
nilpotency class of G is bounded in terms of and the
nilpotency class of by considering Lie rings with a
finite cyclic grading satisfying a certain `selective nilpotency'
condition. The latter technique also yields similar results
bounding the nilpotency class of Lie rings and algebras with a
metacyclic Frobenius group of automorphisms, with corollaries for
connected Lie groups and torsion-free locally nilpotent groups
with such groups of automorphisms. Examples show that such nilpotency results
are no longer true for non-metacyclic Frobenius groups of automorphisms.
© 2014 by Walter de Gruyter Berlin Boston
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Articles in the same Issue
- Masthead
- Pseudovarieties generated by Brauer type monoids
- On some arithmetic properties of Siegel functions (II)
- Perfect vector alignment of the vorticity and the vortex stretching is one forever
- Frobenius groups of automorphisms and their fixed points
- Schubert calculus and the Hopf algebra structures of exceptional Lie groups
- The sovability of norm, bilinear and quadratic equations over finite fields via spectra of graphs
- Periodicity in the stable representation theory of crystallographic groups
- Sums of Fourier coefficients of a Maass form for SL3(ℤ) twisted by exponential functions
- The rational classification of links of codimension > 2
- On the classifying space for proper actions of groups with cyclic torsion
- Turán determinants of Bessel functions
