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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]

Motivated by a classic result for free groups, one says that a group G has the Magnus property if the following holds: Whenever two elements generate the same normal subgroup of G, they are conjugate or inverse-conjugate in G. It is a natural problem to find out which relatively free groups display the Magnus property. We prove that a free polynilpotent group of any given class row has the Magnus property if and only if it is nilpotent of class at most 2. For this purpose we explore the Magnus property more generally in soluble groups, and we produce new techniques, both for establishing and for disproving the property. We also prove that a free center-by-(polynilpotent of given class row) group has the Magnus property if and only if it is nilpotent of class at most 2. On the way, we display 2-generated nilpotent groups (with non-trivial torsion) of any prescribed nilpotency class with the Magnus property. Similar examples of finitely generated, torsion-free nilpotent groups are hard to come by, but we construct a 4-generated, torsion-free, class-3 nilpotent group of Hirsch length 9 with the Magnus property. Furthermore, using a weak variant of the Magnus property and an ultraproduct construction, we establish the existence of metabelian, torsion-free, nilpotent groups of any prescribed nilpotency class with the Magnus property. [ABSTRACT FROM AUTHOR]

Let C 2 ≀ H n denote the wreath product of a cyclic group C 2 of order 2 with a subgroup H n of the symmetric group S n on n letters, and let A be the direct product of a cyclic group of order 2 u with u ≥ 1 and a cyclic group of order 2 v with u ≥ v ≥ 0 . The number of homomorphisms from A to C 2 ≀ H n is denoted by h (A , C 2 ≀ H n) . Let A n be the alternating group on n letters. When H n = A n , h (A , C 2 ≀ A n) has some 2-adic properties which are similar to those of h (A , C 2 ≀ S n) . The exponent of 2 in the decomposition of h (A , C 2 ≀ H n) into prime factors is denoted by ord2 (h (A , C 2 ≀ H n)) . Let [ x ] denote the largest integer not exceeding a real number x . For each n , τ ¯ 2 (u , v) (n) denotes ∑ j = 0 u - 1 [ n / 2 j ] + [ n / 2 u + 1 ] - [ n / 2 u + 2 ] if u = v and denotes ∑ j = 0 u - 1 [ n / 2 j ] - (u - v) [ n / 2 u ] if u ≥ v + 1 , which is known to be the lower bound of ord2 (h (A , C 2 ≀ S n)) . There are three types of 2-adic properties of h (A , C 2 ≀ A n) . The lower bound of ord2 (h (A , C 2 ≀ A n)) is τ ¯ 2 (u , v) (n) if u + δ v 0 ≤ v + 2 , and is τ ¯ 2 (u , v) (n) - 1 if u + δ v 0 ≥ v + 3 . Suppose that u + δ v 0 ≥ v + 2 . For any positive odd integer y , ord2 (h (A , C 2 ≀ A 2 u y)) and ord2 (h (A , C 2 ≀ A 2 u y + 1)) are described by certain 2-adic integers. The values { h (A , C 2 ≀ A n) } n = 0 ∞ are explained by certain 2-adic analytic functions. The results are obtained by using an explicit description of the generating function ∑ n = 0 ∞ h (A , C 2 ≀ A n) X n / (2 n n !) . [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]

This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang–Baxter equation, not necessarily involutive, by means of q-cycle sets. We entirely focus on the finite indecomposable ones, among which we especially study the class of simple solutions. In particular, we provide a group-theoretic characterization of these solutions, including their permutation groups. Finally, we deal with some open questions. [ABSTRACT FROM AUTHOR]

We extend work of Berdinsky and Khoussainov ['Cayley automatic representations of wreath products', International Journal of Foundations of Computer Science27(2) (2016), 147–159] to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups. [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]

Let M be a left module over a ring R with identity and let β be a skew-symmetric R-bilinear form on M. The generalized Heisenberg group consists of the set M × M × R = { (x , y , t) : x , y ∈ M , t ∈ R } with group law (x 1 , y 1 , t 1) (x 2 , y 2 , t 2) = (x 1 + x 2 , y 1 + y 2 , t 1 + β (x 1 , y 2) + t 2). Under the assumption of 2 being a unit in R, we prove that the generalized Heisenberg group decomposes into a product of its subset and subgroup, similar to the well-known polar decomposition in linear algebra. This leads to a parametrization of the generalized Heisenberg group that resembles a parametrization of the Lorentz transformation group by relative velocities and space rotations. [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 study the neighborhoods of a typical point Z n visited at n-th step of a random walk, determined by the condition that the transition probabilities stay close to μ ∗ n (Z n) . If such neighborhood contains a ball of radius C n , we say that the random walk has almost invariant transition probabilities. We prove that simple random walks on wreath products of Z with finite groups have almost invariant distributions. A weaker version of almost invariance implies a necessary condition of Ozawa's criterion for the property H FD . We define and study the radius of almost invariance. We estimate this radius for random walks on iterated wreath products and show this radius can be asymptotically strictly smaller than n / L(n), where L(n) denotes the drift function of the random walk. We show that the radius of individual almost invariance of a simple random walk on the wreath product of Z 2 with a finite group is asymptotically strictly larger than n / L(n). Finally, we show the existence of groups such that the radius of almost invariance is smaller than a given function, but remains unbounded. We also discuss possible limiting distribution of ratios of transition probabilities on non almost invariant scales. [ABSTRACT FROM AUTHOR]

Affine structures of a group G (=braces with adjoint group G) are characterized equationally without assuming further invertibility conditions. If G is finite, the construction of affine structures is reduced to affine structures of p, q-groups. The delicate relationship between finite solvable groups and involutive Yang–Baxter groups is further clarified by showing that much of an affine structure is already inherent in the Sylow system of the group. Semidirect products of braces are modified (shifted) in two ways to handle affine structures of semidirect products of groups. As an application, two of Vendramin's conjectures on affine structures of p, q-groups are verified. A further example illustrates what happens beyond semidirect products. [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]

Let G be an extension of ℚ by a direct sum of r copies of ℚ. (1) If G is abelian, then G is a direct sum of r + 1 copies of ℚ and AutG ≅ GL(r + 1, Q); (2) If G is non-abelian, then G is a direct product of an extraspecial ℚ-group E and m copies of ℚ, where E/ζE is a linear space over Q with dimension 2n and m + 2n = r. Furthermore, let AutG′G be the normal subgroup of AutG consisting of all elements of AutG which act trivially on the derived subgroup G′ of G, and AutG/ζG,ζGG be the normal subgroup of AutG consisting of all central automorphisms of G which also act trivially on the center ζG of G. Then (i) The extension 1 → AutG′G → AutG → AutG′ → 1 is split; (ii) AutG′G/AutG/ζG,ζGG ≅ Sp(2n,Q) × (GL(m, Q) ⋉ ℚ(m)); (iii) AutG/ζG,ζGG/InnG ≅ ℚ(2nm). [ABSTRACT FROM AUTHOR]

The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let G be the direct product of a generalized extraspecial ℤ-group E and a free abelian group A with rank m, where E={(1kα1kα2⋯kαnαn+1010⋯0αn+2⋮⋮⋮⋮⋮000⋯1α2n+1000⋯01)|αi∈Z,i=1,2,...,2n+1}, where k is a positive integer. Let AutG′G be the normal subgroup of AutG consisting of all elements of AutG which act trivially on the derived subgroup G′ of G, and AutG/ζG,ζGG be the normal subgroup of AutG consisting of all central automorphisms of G which also act trivially on the center ζG of G. Then (i) The extension 1 → AutG′G → AutG → AutG′ → 1 is split. (ii) AutG′G/AutG/ζG,ζGG ≅ Sp(2n, Z) × (GL(m, Z) × (ℤ)m). (iii) AutG/ζG,ζGG/InnG ≅ (ℤk)2n ⊕ (ℤk)2nm. [ABSTRACT FROM AUTHOR]

We show how to construct all the extensions of left braces by ideals with trivial structure. This is useful to find new examples of left braces. But, to do so, we must know the basic blocks for extensions: the left braces with no ideals besides the trivial and the total ideal, called simple left braces. In this article, we present the first non-trivial examples of finite simple left braces. To explicitly construct them, we define the matched product of two left braces, which is a useful method to recover a finite left brace from its Sylow subgroups. [ABSTRACT FROM AUTHOR]

We explicitly construct infinitely many a non-abelian strongly real Beauville [ABSTRACT FROM AUTHOR]

We prove that one cannot algorithmically decide whether a finitely presented Z -extension admits a finitely generated base group, and we use this fact to prove the undecidability of the BNS invariant. Furthermore, we show the equivalence between the isomorphism problem within the subclass of unique Z -extensions, and the semi-conjugacy problem for deranged outer automorphisms. [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]

We investigate the recently obtained condition [] for amenability of groups generated by bounded automorphisms of a spherically homogeneous rooted tree together with our group construction [] based on the notion of a homogeneous automorphism and its crack. We establish the general condition answering the question when a group generated by homogeneous automorphisms and their cracks is amenable. We apply this result to the universal construction from [], providing finitely generated, amenable groups, which are dense in infinitely iterated, permutational wreath products of transitive, Abelian 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]

Inspired by the definition of generalized wreath product of permutation groups, we define the generalized wreath product of graphs, containing the classical Cartesian and wreath product of graphs as particular cases. We prove that the generalized wreath product of Cayley graphs of finite groups is the Cayley graph of the generalized wreath product of the corresponding groups. [ABSTRACT FROM AUTHOR]

We study free centre-by-(abelian-by-exponent 2) groups. Our main result is a complete description of the centre. It is isomorphic to a direct sum of a free abelian group and a torsion subgroup. The latter is a direct sum of cyclic groups of order two and cyclic groups of order four. We exhibit a generating set consisting of elements of infinite order, order 2, and order 4, such that the centre is the direct sum of cyclic subgroups generated by those generators. Our approach makes essential use of homological methods. [ABSTRACT FROM AUTHOR]

In this paper we are interested in investigating the principal series representations of a certain rank 2 Kac-Moody group. This group (denoted by ) is introduced as a central extension by of the group of two-by-two unimodular matrices over the non-archimedean local field, . We begin by determining a formula for a cocycle that defines this central extension. This is used to classify all irreducible representations of the centrally extended torus inside and, as a consequence of this classification, we are able to determine the irreducible unramified principal series representations of . [ABSTRACT FROM AUTHOR]

We use the combinatorial language of automata to define and study profinite groups which are infinitely iterated permutational wreath products of transitive finite permutation groups. We provide some naturally defined automaton realizations of these groups by the so-called time-varying automata, as well as by Mealy automata to characterize these groups as automaton groups, i.e. as groups topologically generated by a single automaton. [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 exhibit a finitely generated group G and a sequence of finite index normal subgroups [InlineMediaObject not available: see fulltext.] such that for every finite generating subset $${S\subseteq G}$$ , the sequence of finite Cayley graphs ( G/ N, S) does not coarsely embed into any L-space for $${1 \leqslant p < \infty}$$ (moreover, into any uniformly curved Banach space), and yet admits no weakly embedded expander. The reason why our examples do not coarsely embed is a new phenomenon called relative expansion, which we define in terms of Poincaré inequalities. [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]