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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]

In Straubing (Theor. Comput. Sci. 13:137–150, 1981), Straubing introduced an n-ary the Schützenberger product on finite monoids. In this paper, we examine this generalization from another point of view and obtain an alternative n-generalized monoid construction of Schützenberger product to facilitate the intelligibility of Straubing's structure. Then, we find a generator and relator set for this product. Additionally, we compute a complete rewriting system and thus we obtain normal form structure of elements of this monoid construction which could be useful for Automata Theory. This gives us the solvability of the word problem for this product. [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]

In the paper, we develop further the properties of Schur rings over infinite groups, with particular emphasis on the virtually cyclic group Z × Z p , where p is a prime. We provide structure theorems for primitive sets in these Schur rings. In the case of Fermat and safe primes, a complete classification theorem is proven, which states that all Schur rings over Z × Z p are traditional. We also draw analogs between Schur rings over Z × Z p and partitions of difference sets over Z p . [ABSTRACT FROM AUTHOR]

Let N and H be groups, and let G be an extension of H by N. In this article, we describe the structure of the complex group ring of G in terms of data associated with N and H. In particular, we present conditions on the building blocks N and H guaranteeing that G satisfies the zero-divisor and idempotent conjectures. Moreover, for central extensions involving amenable groups we present conditions on the building blocks guaranteeing that the Kadison–Kaplansky conjecture holds for the group C ∗ -algebra of G. [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]

Rota–Baxter operators for groups were recently introduced by Guo, Lang, and Sheng. Bardakov and Gubarev showed that with each Rota–Baxter operator, one can associate a skew brace. Skew braces on a group G can be characterised in terms of certain gamma functions from G to its automorphism group Aut (G) that are defined by a functional equation. For the skew braces obtained from a Rota–Baxter operator, the corresponding gamma functions take values in the inner automorphism group Inn (G) of G. In this paper, we give a characterisation of the gamma functions on a group G, with values in Inn (G) , that come from a Rota–Baxter operator, in terms of the vanishing of a certain element in a suitable second cohomology group. Exploiting this characterisation, we are able to exhibit examples of skew braces whose corresponding gamma functions take values in the inner automorphism group, but cannot be obtained from a Rota–Baxter operator. For gamma functions that can be obtained from a Rota–Baxter operators, we show how to get the latter from the former, exploiting the knowledge that a suitable central group extension splits. [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]

Employing the algebraic structure of the left brace and the dynamical extensions of cycle sets, we investigate a class of indecomposable involutive set-theoretic solutions of the Yang–Baxter equation having specific imprimitivity blocks. Moreover, we study one-generator left braces of multipermutation level 2. [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]

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$ , where $C$ is a finite group and $F$ a non-abelian free group. [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]

Let G be a group. The orbits of the natural action of Aut (G) on G are called automorphism orbits of G, and the number of automorphism orbits of G is denoted by ω (G) . Let G be a virtually nilpotent group such that ω (G) < ∞ . We prove that G = K ⋊ H where H is a torsion subgroup and K is a torsion-free nilpotent radicable characteristic subgroup of G. Moreover, we prove that G ′ = D × Tor (G ′) where D is a torsion-free nilpotent radicable characteristic subgroup. In particular, if the maximum normal torsion subgroup τ (G) of G is trivial, then G ′ is nilpotent. [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]

Let G be a group. The orbits of the natural action of Aut (G) on G are called "automorphism orbits" of G, and the number of automorphism orbits of G is denoted by ω (G) . We prove that if G is a soluble group of finite rank such that ω (G) < ∞ , then G contains a torsion-free radicable nilpotent characteristic subgroup K such that G = K ⋊ H , where H is a finite group. Moreover, we classify the mixed order soluble groups of finite rank such that ω (G) = 3 . [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]