Nil-automorphisms of groups with residual properties

Let G be any group and x an automorphism of G. The automorphism x is said to be nil if, for every g ∈ G, there exists n = n(g) such that [g, _n x] = 1. If n can be chosen independently of g, we say that x is n-unipotent. A nil (resp. unipotent) automorphism x could also be seen as a left Engel element (resp. left n-Engel element) in the group G〈x〉. When G is a finite dimensional vector space, groups of unipotent linear automorphisms turn out to be nilpotent, so that one might ask to what extent this result can be extended to a more general setting. In this paper we study finitely generated groups of nil or unipotent automorphisms of groups with residual properties (e.g. locally graded groups, residually finite groups, profinite groups), proving that such groups are nilpotent.     

Extensions of Chevalley groups

LetG ₀ be a split simple Chevalley group of any type over the fieldK andG its universal group; and let? ₀ be the group of automorphisms of the corresponding Chevalley algebra,L _K, generated byG ₀ and all the diagonal automorphisms. A group? (and appropriate homorphisms) is constructed which generalizes the groupGL _n (K) whenG ₀ is specialized to typeA _n?1.     

Contractible Lie groups over local fields

Let G be a Lie group over a local field of characteristic p > 0 which admits a contractive automorphism α : G → G (i.e., α ⁿ (x) → 1 as n → ∞, for each x ∈ G). We show that G is a torsion group of finite exponent and nilpotent. We also obtain results concerning the interplay between contractive automorphisms of Lie groups over local fields, contractive automorphisms of their Lie algebras, and positive gradations thereon. Some of the results extend to Lie groups over arbitrary complete ultrametric fields. Supported by the German Research Foundation (DFG), grants GL 357/2-1 and GL 357/6-1.     

The hypercentral structure of the group of unitriangular automorphisms of a free metabelian Lie algebra

We describe the hypercentral structure of the group of unitriangular automorphisms of a free metabelian Lie algebra over an arbitrary field. Using it, we prove that this group admits no faithful representation by matrices over a field provided that the algebra rank is at least four.     

Adian-Lisenok groups and (U) condition

A group G possesses the property (U) with respect to S if there exists a number M = M(G) such that for each generating set P of the group G there exists an element t ? G for which max _x?S |t ^?1 xt| _P ≤ M. It is proved that the well-known Adian-Lisenok groups possess the property (U). In connection with the problem on finding infinite groups with the property (U), which is stated in a joint unpublishedwork by D.Osin and D. Sonkin, it is shown that for any odd n ≥ 1003 there is a continuum set of non-isomorphic, i.e. simple groups with the property (U) in the variety of groups satisfying the identity x ⁿ = 1.     

Doubly transitive automorphism groups of block designs

If G is a doubly transitive group of automorphisms of a block design with λ = 1, then for any block Δ of the design and any point α in Δ, the set Δ?{α} is a block of imprimitivity for G_α. What are sufficient conditions for a doubly transitive but not doubly primitive permutation group G to be a group of automorphisms of a non-trivial block design with λ = 1 ? Can the design or the group G be identified if there is a nonidentity automorphism in G fixing every point of some block of the design? Both of these questions are investigated and some answers are given.     

The conjugacy problem for Dehn twist automorphisms of free groups

A Dehn twist automorphism of a group G is an automorphism which can be given (as specified below) in terms of a graph-of-groups decomposition of G with infinite cyclic edge groups. The classic example is that of an automorphism of the fundamental group of a surface which is induced by a Dehn twist homeomorphism of the surface. For , a non-abelian free group of finite rank n, a normal form for Dehn twist is developed, and it is shown that this can be used to solve the conjugacy problem for Dehn twist automorphisms of . Received: February 12, 1996.     

Normalizers of Chevalley Groups of Type G 2 Over Local Rings Without 1/2

In this paper, we prove that every element of the linear group GL₁₄(R) normalizing the Chevalley group of type G ₂ over a commutative local ring R without 1/2 belongs to this group up to some multiplier. This allows us to improve our classification of automorphisms of these Chevalley groups showing that an automorphism-conjugation can be replaced by an inner automorphism. Therefore, it is proved that every automorphism of a Chevalley group of type G ₂ over a local ring without 1/2 is a composition of a ring and an inner automorphisms.     

Ordering groups and validity in lattice-ordered groups

An inductive characterization is given of the subsets of a group that extend to the positive cone of a right order on the group. This characterization is used to relate validity of equations in lattice-ordered groups (?-groups) to subsets of free groups that extend to the positive cone of a right order. As a consequence, new proofs are obtained of the decidability of the word problem for free ?-groups and generation of the variety of ?-groups by the ?-group of automorphisms of the real line. An inductive characterization is also given of the subsets of a group that extend to the positive cone of an order on the group. In this case, the characterization is used to relate validity of equations in varieties of representable ?-groups to subsets of relatively free groups that extend to the positive cone of an order.     

Splitting automorphisms of order p k of free Burnside groups are inner

It is proved that, if the order of a splitting automorphism of odd period n ≥ 1003 of a free Burnside group B(m, n) is equal to a power of some prime, then the automorphism is inner. Thus, an affirmative answer is given to the question concerning the coincidence of the splitting automorphisms of the group B(m, n) with the inner automorphisms for all automorphisms of order p ^k (p is a prime). This question was posed in 1990 in “Kourovka Notebook” (see the 11th edition, Question 11.36.b).     

Polycyclic group admitting an almost regular automorphism of prime order

A well-known result due to Thompson states that if a finite group G has a fixed-point-free automorphism of prime order, then G is nilpotent. In this note, giving a counterpart of Thompson's result in the context of polycyclic groups, we prove: if a polycyclic group G has an automorphism of prime order with finitely many fixed points, then G is nilpotent-by-finite.     

On automorphisms and embeddings of free periodic groups

The paper gives a construction of a free monoid of rank 2 in the group of automorphisms of free periodic groups B(m, n) of any odd period n ≥ 665 and any rank m > 1.Moreover, it is proved that if the period is any prime numbern > 1003 and the group B(m, n) is nested in some n-periodic group G as a normal subgroup, then B(m, n) is a direct factor in G.     

Growth of connected locally compact groups

The asymptotic behavior of the Haar measure of Uⁿ = U · U ··· U, where U is a compact neighborhood of the identity in a separable, connected locally compact group, is considered. It is shown that for a given group G, the measure of Uⁿ grows at most polynomially for all U ? G, or at least exponentially for all U ? G. The groups having polynomial growth are characterized in terms of uniformly discrete free subsemigroups and in terms of the eigenvalues of elements occurring in the adjoint group of an approximating Lie group.     

Transitivity degrees of countable groups and acylindrical hyperbolicity

We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive faithful actions of mapping class groups. It also implies that in various geometric and algebraic settings, the transitivity degree of an infinite group can only take two values, namely 1 and ∞. Here, by transitivity degree of a group we mean the supremum of transitivity degrees of its faithful permutation representations. Further, for any countable group G admitting a highly transitive faithful action, we prove the following dichotomy: Either G contains a normal subgroup isomorphic to the infinite alternating group or G resembles a free product from the model theoretic point of view. We apply this theorem to obtain new results about universal theory and mixed identities of acylindrically hyperbolic groups. Finally, we discuss some open problems.     

Embedding paratopological groups into topological products

We show that a Hausdorff paratopological group G admits a topological embedding as a subgroup into a topological product of Hausdorff first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the Hausdorff number of G is countable, i.e., for every neighbourhood U of the neutral element e of G there exists a countable family γ of neighbourhoods of e such that _?V∈γVV−1⊆U. Similarly, we prove that a regular paratopological group G can be topologically embedded as a subgroup into a topological product of regular first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the index of regularity of G is countable.As a by-product, we show that a regular totally ω-narrow paratopological group with countable index of regularity is Tychonoff.     

Primitivity in the free groups of the variety UmUn

For each m,n >= 0, let G_m,n denote the free group of rank r in the variety U_mU_n. The main results in this paper are: (i) a necessary and sufficient condition for a system of r elements {v₁,…, v_r} to form a basis of G_m,ni (ii) necessary and sufficient conditions for a system of l elements {v₁,…, v_l}, l <= r, to be included in a basis of G_m,0. In particular, (i), (ii) yield corresponding results for the free metabelian group of rank r.     

Weakly power automorphisms of groups

An automorphism α of a group G is called a weakly power automorphism if it maps every non-periodic subgroup of G onto itself. The aim of this paper is to investigate the behavior of weakly power automorphisms. In particular, among other results, it is proved that all weakly power automorphisms of a soluble non-periodic group G of derived length at most 3 are power automorphisms, i.e. they fix all subgroups of G. This result is best possible, as there exists a soluble non-periodic group of derived length 4 admitting a weakly power automorphism, which is not a power automorphism.     

A counter example for nilpotent groups

Let R be a ring with 1, I be a nilpotent subring of R (there exists a natural number n, such that In = 0), and I be generated by {xj |j ∈ J} as ring. Write U = 1 + I, and it is a nilpotent group with class ≤ n - 1. Let G be the subgroup of U which is generated by {1 + xj|j ∈ J}. The group constructed in this paper indicates that the nilpotency class of G can be less than that of U.     

Automorphism groups of finite posets

For any finite group G, we construct a finite poset (or equivalently, a finite T₀-space) X, whose group of automorphisms is isomorphic to G. If the order of the group is n and it has r generators, X has n(r+2) points. This construction improves previous results by G. Birkhoff and M.C. Thornton. The relationship between automorphisms and homotopy types is also analyzed.