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 2n2^n

Suppose that a finite group $G$ admits an automorphism $\varphi$ of order $2^n$ such that the fixed-point subgroup $C_G(\varphi ^{2^{n-1}})$ of the involution $\varphi ^{2^{n-1}}$ is nilpotent of class $c$. Let $m=|C_G(\varphi)|$ be the number of fixed points of $\varphi$. It is proved that $G$ has a characteristic soluble subgroup of derived length bounded in terms of $n,c$ whose index is bounded in terms of $m,n,c$. A similar result is also proved for Lie rings.Comment: minor corrections and addition