An Engel condition for orderable groups

Let m,n be positive integers, v a multilinear commutator word and w=v^m. We prove that if G is an orderable group in which all w-values are n-Engel, then the verbal subgroup v(G) is locally nilpotent. We also show that in the particular case where v=x the group G is nilpotent (rather than merely locally nilpotent)

Words of Engel type are concise in residually finite groups

Given a group-word w and a group G, the verbal subgroup w(G) is the one generated by all w-values in G. The word w is said to be concise if w(G) is finite whenever the set of w-values in G is finite. In the sixties P. Hall asked whether every word is concise but later Ivanov answered this question in the negative. On the other hand, Hall\u2019s question remains wide open in the class of residually finite groups. In the present article we show that various generalizations of the Engel word are concise in residually finite groups

Derived Subgroups of Fixed Points in Profinite Groups

The main result of this paper is the following theorem. Let q be a prime, A an elementary abelian group of order q^3. Suppose that A acts as a coprime group of automorphisms on a profinite group G in such a manner that C_G(a)' is periodic for each nontrivial element a in A. Then G' is locally finite.Comment: To appear in Glasgow Mathematical Journal (2011). 11 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

On the length of finite factorized groups

The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of non-abelian simple groups. The generalized Fitting height of a finite group G is the least number h=h ∗ (G) such that F ∗ h (G)=G , where F ∗ 1 (G)=F ∗ (G) is the generalized Fitting subgroup, and F ∗ i+1 (G) is the inverse image of F ∗ (G/F ∗ i (G)) . It is proved that if a finite group G=AB is factorized by two subgroups of coprime orders, then the nonsoluble length of G is bounded in terms of the generalized Fitting heights of A and B . It is also proved that if, say, B is soluble of derived length d , then the generalized Fitting height of G is bounded in terms of d and the generalized Fitting height of A