2,254 research outputs found
Finite p-groups with a Frobenius group of automorphisms whose kernel is a cyclic p-group
Suppose that a finite p-group G admits a Frobenius group of automorphisms
FH with kernel F that is a cyclic p-group and with complement H. It is proved
that if the fixed-point subgroup CG(H) of the complement is nilpotent of class c,
then G has a characteristic subgroup of index bounded in terms of c, jCG(F)j, and
jFj whose nilpotency class is bounded in terms of c and jHj only. Examples show
that the condition of F being cyclic is essential. The proof is based on a Lie ring
method and a theorem of the authors and P. Shumyatsky about Lie rings with a
metacyclic Frobenius group of automorphisms FH. It is also proved that G has a
characteristic subgroup of (jCG(F)j; jFj)-bounded index whose order and rank are
bounded in terms of jHj and the order and rank of CG(H), respectively, and whose
exponent is bounded in terms of the exponent of CG(H)
Accelerating cosmologies in Lovelock gravity with dilaton
For the description of the Universe expansion, compatible with observational
data, a model of modified gravity - Lovelock gravity with dilaton - is
investigated. D-dimensional space with 3- and (D-4)-dimensional maximally
symmetric subspaces is considered. Space without matter and space with perfect
fluid are under test. In various forms of the theory under way (third order
without dilaton and second order - Einstein-Gauss-Bonnet gravity - with dilaton
and without it) stationary, power-law, exponential and exponent-of-exponent
form cosmological solutions are obtained. Last two forms include solutions
which are clear to describe accelerating expansion of 3-dimensional subspace.
Also there is a set of solutions describing cosmological expansion which does
not tend to isotropization in the presence of matter.Comment: 23 page
Finite groups and Lie rings with an automorphism of order
Suppose that a finite group admits an automorphism of order
such that the fixed-point subgroup of the
involution is nilpotent of class . Let
be the number of fixed points of . It is proved
that has a characteristic soluble subgroup of derived length bounded in
terms of whose index is bounded in terms of . A similar result is
also proved for Lie rings.Comment: minor corrections and addition
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