Engel Values in Residually Finite Groups

The following theorem is proved. Let n be a positive integer and q a power of a prime p. There exists a number m = m(n, q) depending only on n and q such that if G is any residually finite group satisfying the identity ([x _1,n y ₁] ⋯ [x _m,n y _m ])^q ≡ 1, then the verbal subgroup of G corresponding to the nth Engel word is locally finite.     

On periodic solvable groups having automorphisms with nilpotent fixed point groups

Letp be a prime,G a periodic solvablep′-group acted on by an elementary groupV of orderp ². We show that ifC _G(v) is abelian for eachv ∈V ^# thenG has nilpotent derived group, and ifp=2 andC _G(v) is nilpotent for eachv ∈V ^# thenG is metanilpotent. Earlier results of this kind were known only for finite groups.     

Exponent of a finite group of odd order with an involutory automorphism

Let $$w = w(x_1, \ldots , x_n)$$ be a non-trivial word of n-variables. The word map on a group G that corresponds to w is the map $$\widetilde{w}: G^n\rightarrow G$$ where $$\widetilde{w}((g_1, \ldots , g_n)) := w(g_1, \ldots , g_n)$$ for every sequence $$(g_1, \ldots , g_n)$$ . Let $$\mathcal G$$ be a simple and simply connected group which is defined and split over an infinite field K and let $$G =\mathcal G(K)$$ . For the case when $$w = w_1w_2 w_3 w_4$$ and $$w_1, w_2, w_3, w_4$$ are non-trivial words with independent variables, it has been proved by Hui et al. (Israel J Math 210:81–100, 2015) that $$G{\setminus } Z(G) \subset {{\text { Im}}}\,\widetilde{w}$$ where Z(G) is the center of the group G and $${{\text { Im}}}\, {\widetilde{w}}$$ is the image of the word map $$\widetilde{w}$$ . For the case when $$G = {{\text {SL}}}_n(K)$$ and $$n \ge 3$$ , in the same paper of Hui et al. (2015) it was shown that the inclusion $$G{\setminus } Z(G)\subset {{\text { Im}}}\,\widetilde{w}$$ holds for a product $$w = w_1w_2 w_3$$ of any three non-trivial words $$ w_1, w_2, w_3$$ with independent variables. Here we extent the latter result for every simple and simply connected group which is defined and split over an infinite field K except the groups of types $$B_2, G_2$$ .     

Engel words and the Restricted Burnside Problem

The following theorem is proved. For any positive integers n and k there exists a number s = s(n, k) depending only on n and k such that the class of all groups G satisfying the identity ^n 1{\left(\left[x_1, {}_ky_1\right] \cdots \left[x_s, {}_ky_s\right]\right)^n \equiv 1} and having the verbal subgroup corresponding to the kth Engel word locally finite is a variety.     

On locally finite groups with a four-subgroup whose centralizer is small

Let $G$ be a locally finite group which contains a non-cyclic subgroup $V$ of order four such that $C_{G}\left( V\right) $ is finite and $C_{G}\left( \phi \right)$ has finite exponent for some $\phi \in V$ . We show that $[G,\phi ]^{\prime }$ has finite exponent. This enables us to deduce that $G$ has a normal series $1\le G_1\le G_2\le G_3\le G$ such that $G_1$ and $G/G_2$ have finite exponents while $G_2/G_1$ is abelian. Moreover $G_3$ is hyperabelian and has finite index in $G$ .     

Nilpotent Ideals in Graded Lie Algebras and Almost Constant-Free Derivations

It is proved that if a (?/p ?)-graded Lie algebra L, where p is a prime, has exactly d nontrivial grading components and dim L ₀ = m, then L has a nilpotent ideal of d-bounded nilpotency class and of finite (m,d)-bounded codimension. As a consequence, Jacobson's theorem on constant-free nilpotent Lie algebras of derivations is generalized to the almost constant-free case. Another application is for Lie algebras with almost fixed-point-free automorphisms.