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In this paper we make explicit an application of the wreath product construction to the Galois groups of field extensions. More precisely, given a tower of fields F ⊆ K ⊆ L with L/F finite and separable, we explicitly construct an embedding of the Galois group Gal (L c / F) into the regular wreath product Gal (L c / K c) ≀ r Gal (K c / F) . Here Lc (resp. Kc) denotes the Galois closure of L/F (resp. K/F). Similarly, we also construct an explicit embedding of the Galois group Gal (L c / F) into the smaller sized wreath product Gal (L c / K) ≀ Ω Gal (K c / F) , where Ω = Hom F (K , K c) is acted on by composition of automorphisms in Gal (K c / F) . Moreover, when L/K is a Kummer extension we prove a sharper embedding, that is, that Gal (L c / F) embeds into the wreath product Gal (L / K) ≀ Ω Gal (K c / F) . As corollaries we obtain embedding theorems when L/K is cyclic and when it is quadratic with char (F) ≠ 2 . We also provide examples of these embeddings and as an illustration of the usefulness of these embedding theorems, we survey some recent applications of these types of results in field theory, arithmetic statistics, number theory and arithmetic geometry. [ABSTRACT FROM AUTHOR]

A graph Γ is said to be 2-arc-transitive if its automorphism group acts transitively on the set of 2-arcs of Γ . In this paper, we give a group-theoretic characterization of those connected 2-arc-transitive graphs which admit a vertex-transitive simple group. [ABSTRACT FROM AUTHOR]

AbstractWe find an explicit form of the inverse isomorphism from Shapiro’s lemma in terms of inhomogeneous cocycles and apply it to construct special nonsplit coverings of groups with a unique conjugacy class of involutions. [ABSTRACT FROM AUTHOR]

Let F be a non-cyclic free group of rank n. Consider the quotient F / [ F ″ , F ] , the free centre-by-metabelian group of rank n. In 1973, C. K. Gupta proved by purely group theoretic means that it contains an elementary abelian 2-group of rank ( 4 n) in its centre for n ≥ 4 , and exhibited an explicit generating set for this torsion subgroup. In this paper, using homological methods, we provide an alternative explicit generating set for it, and identify this torsion subgroup as the isolator of an explicitly given subgroup in the quotient F ″ / [ F ″ , F ′ ] . [ABSTRACT FROM AUTHOR]

Given a group G and n ≥ 0 , let W(G, n) be the associated iterated wreath product—unrestricted when G is infinite—viewed as a permutation group on Gn. We prove that the normalizer of W(G, n) in the symmetric group S (G n) is equal to M n ⋉ W (G , n) , where Mn is isomorphic to Aut (G) n . The action of Aut (G) n on W(G, n) is recursively described. Communicated by Mark Lewis [ABSTRACT FROM AUTHOR]

We establish a connection between the generalized conjugacy problem for a G-by- Z group, GCP (G ⋊ Z) , and two algorithmic problems for G: the generalized Brinkmann's conjugacy problem, GBrCP(G), and the generalized twisted conjugacy problem, GTCP(G). We explore this connection for generalizations of different kinds: relative to finitely generated subgroups, to their cosets, or to recognizable, rational, context-free or algebraic subsets of the group. Using this result, we are able to prove that GBrCP(G) is decidable (with respect to cosets) when G is a virtually polycyclic group, which implies in particular that the generalized Brinkmann's equality problem, GBrP(G), is decidable if G is a finitely generated abelian group. Finally, we prove that if G is a finitely generated virtually free group, then the simple versions of Brinkmann's equality problem and of the twisted conjugacy problem, BrP(G) and TCP(G), are decidable. [ABSTRACT FROM AUTHOR]

The celebrated Kaloujnine-Krasner theorem associates, with a short exact sequence 1 → N → ι G → π H → 1 of groups and a section s : H → G , an embedding Φ : G → N ≀ H of G into the (unrestricted) wreath product of N and H. Given two groups H and N, a short exact sequence as above is called an extension of H by N, denoted by (G ; ι , π) . Moreover, one says that two extensions (G 1 ; ι 1 , π 1) and (G 2 ; ι 2 , π 2) of H by N are equivalent if there exists a group isomorphism η : G 1 → G 2 such that ι 2 = η ° ι 1 and π 1 = π 2 ° η . We say that two embeddings Φ 1 : G 1 → N ≀ H and Φ 2 : G 2 → N ≀ H are equivalent if there exists a group isomorphism η : G 1 → G 2 such that Φ 1 = Φ 2 ° η . We show that two extensions (G 1 ; ι 1 , π 1) and (G 2 ; ι 2 , π 2) are equivalent if and only if the embeddings Φ 1 and Φ 2 , associated with any two sections s 1 : H → G 1 and s 2 : H → G 2 via the Kaloujnine-Krasner theorem, are equivalent. [ABSTRACT FROM AUTHOR]

Let F be a free group of rank d ≥ 2. Consider the quotient F / [ γ 4 (F ′) , F ] F ″ ′ , the free centre-by-(nilpotent of class 3)-by-abelian group within the variety of all soluble groups of derived length 3. A peculiar feature of this group is the occurence of torsion in its centre. Our main result is a complete description of an elementary abelian 2-group of rank ( d 4 ) in F / [ γ 4 (F ′) , F ] F ″ ′ in terms of generators of F for d ≥ 4. Our approach makes essential use of homological methods. [ABSTRACT FROM AUTHOR]

Motivated by the reduction techniques involving character triples for the local-global conjectures, we show that a blockwise relation between module triples is a consequence of a derived equivalence with additional properties. Moreover, we show that this relation is compatible with wreath products. [ABSTRACT FROM AUTHOR]

This is one of a series of papers which aim toward determining the automorphism groups of Frobenius groups. This paper solves the problem in the case where the Frobenius kernels are elementary abelian and Frobenius complements are cyclic. [ABSTRACT FROM AUTHOR]

We develop a method to construct all the indecomposable involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation with a prime-power number of elements and cyclic permutation group. Moreover, we give a complete classification of the indecomposable ones having abelian permutation group and cardinality pq (where p and q are prime numbers not necessarily distinct). [ABSTRACT FROM AUTHOR]

We prove that for any automorphism of the restricted wreath product and the Reidemeister number is infinite (the property). For and , where p > 3 is prime, we give examples of automorphisms with finite Reidemeister numbers. So these groups do not have the property. For these groups and , where m is relatively prime to 6, we prove the twisted Burnside-Frobenius theorem (TBFTf): if , then it is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action. [ABSTRACT FROM AUTHOR]

Letbe a cyclic group of orderpm, wherepis a prime andmis an arbitrary positive integer. It is proved that the wreath product(nfactors) is determined for each naturalnby its endomorphism semigroup in the class of all groups. It follows that (a) every Sylow subgroup of a finite symmetric group is determined by its endomorphism semigroup in the class of all groups, (b) each finitep-groupGis embeddable into a finitep-groupG* such thatG* is determined by its endomorphism semigroup in the class of all groups. [ABSTRACT FROM AUTHOR]

In this paper, we construct two self-similar groups factorizable by their locally finite 2-subgroups, generators of which are constructed from the generators of the first Grigorchuk group. Both groups contain the first Grigorchuk group, the second also contains the binary adding machine. The first group is a 2-group but not locally finite and the second group is neither locally finite nor a 2-group. Therefore each of them is a new example providing a negative answer to well-known solved problems. Surprisingly, definitions of both groups involve the Gray code. [ABSTRACT FROM PUBLISHER]

Let 𝒞 be a class of groups. We give sufficient conditions ensuring that a free product of residually 𝒞 groups is again residually 𝒞, and analogous conditions are given for LE-𝒞 groups. As a corollary, we obtain that the class of residually amenable groups and the one of locally embeddable into amenable (LEA) groups are closed under taking free products. Moreover, we consider the pro-𝒞 topology and we characterize special HNN extensions and amalgamated free products that are residually 𝒞, where 𝒞 is a suitable class of groups. In this way, we describe special HNN extensions and amalgamated free products that are residually amenable. [ABSTRACT FROM PUBLISHER]

Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behavior of conjugacy length functions under group extensions, introducing the twisted and restricted conjugacy length functions. We apply these results to show that certain abelian-by-cyclic groups have linear conjugacy length function and certain semidirect products ℤd ⋊ ℤkhave at most exponential (ifk > 1) or linear (ifk = 1) conjugacy length functions. [ABSTRACT FROM AUTHOR]

We investigate the palindromic width of finitely generated solvable groups. We prove that every finitely generated 3-step solvable group has finite palindromic width. More generally, we show the finiteness of the palindromic width for finitely generated abelianby- nilpotent-by-nilpotent groups. For arbitrary solvable groups of step > 3, we prove I that if G is a finitely generated solvable group that is an extension of an abelian group by a group satisfying the maximal condition for norma! subgroups, then the palindromic j width of G is finite. We also prove that the palindromic width of Z:Z with respect to the set of standard generators is 3. [ABSTRACT FROM AUTHOR]

We introduce a class of function spaces consisting of integer-valued functions of several integers which coordinatize the elements of certain subgroups of some finite regular wreath product groups. On each function space, we define operators which correspond to forming certain commutators relevent to computing the upper central series. We define an automorphism of the function space which enables us to define a class of subgroups that is useful for describing the upper central series of certain finite regular wreath productp-groups. Our results describe fundamental, interesting, and useful relationships between the automorphism and the operators. We describe some applications. [ABSTRACT FROM AUTHOR]

We study profinite groups which are infinitely iterated wreath productsW∞= …≀Cn 2≀Cn 1of finite cyclic groups via combinatorial language of transducers. Namely, we provide a naturally defined automaton realization of the groupW∞by an automaton over a changing alphabet. Our construction gives a characterization of these profinite groups as automaton groups, i.e. as groups generated by a single automaton. [ABSTRACT FROM PUBLISHER]

Let (n1, n2,...) be an infinite sequence of positive integers greater than 1. We prove that the topological rank of the corresponding infinitely iterated wreath product ...wrCn2wrCn1 of finite cyclic groups is equal to k + 1, where k denotes the topological rank of the direct product Cn2 × Cn3 × ... of finite cyclic groups (this rank may be of finite or infinite cardinality, that is, k is a finite number or k = ∞). We construct the minimal set generating topologically the above wreath product as a set of the so-called cyclic automorphisms of a spherically homogeneous rooted tree. [ABSTRACT FROM AUTHOR]

Let W be a first-order structure and ρ be an Aut(W)-congruence on W. In this article we define the almost-free finite covers of W with respect to ρ, and we show how to construct them. These are a generalization of free finite covers. A consequence of a result of [5] is that any finite cover of W with binding groups all equal to a simple non-abelian permutation group is almost-free with respect to some ρ on W. Our main result gives a description (up to isomorphism) in terms of the Aut(W)-congruences on W of the kernels of principal finite covers of W with binding groups equal at any point to a simple non-abelian regular permutation group G. Then we analyze almost-free finite covers of Ω(n), the set of ordered n-tuples of distinct elements from a countable set Ω, regarded as a structure with Aut(Ω(n)) = Sym(Ω) and we show a result about biinterpretability. The material here presented addresses a problem which arises in the context of classification of totally categorical structures. [ABSTRACT FROM AUTHOR]

It is known that for any finitely generated group G from the large class of “locally graded” groups, satisfaction of an Engel or positive law forces G to be virtually nilpotent. Black (1999) gives a sufficient condition for an arbitrary 2-variable law w(x, y)≡ 1 to imply virtual nilpotence—though only for finitely generated residually finite groups. We show how the Dichotomy Theorem from Burns and Medvedev (2003) for arbitrary words w(x1,...,xn), encompasses Black's condition, extending it to the n-variable case and a certain large class S (however still falling short of the class of locally graded groups). We infer in particular that her condition is also necessary. We also deduce a simplified version of an algorithm of Li (2004, 2005) for deciding whether or not a given law w(x1,...,xn)≡ 1 satisfies the extended version of Black's criterion. [ABSTRACT FROM AUTHOR]

Assume P is a family of primes, and let ()P represent the P-localization functor. If is a group extension giving rise to a nilpotent action of G on N, we prove that the sequence is exact. Moreover, in the case where Q satisfies a certain pair of homological conditions, we show that the map ιP is an injection. This generalizes the well-known result that ()P is exact in the category of nilpotent groups. Applications are given to calculating P-localizations of virtually nilpotent groups. [ABSTRACT FROM AUTHOR]