Search

A generalization to the categorical notion of biproduct, called semibiproduct, which in the case of groups covers classical semidirect products, has recently been analysed in the category of monoids with surprising results in the classification of weakly Schreier extensions. The purpose of this paper is to extend the study of semibiproducts to the category of semigroups. However, it is observed that a further analysis into the category of magmas is required in attaining a full comprehension on the subject. Indeed, although there is a subclass of magma-actions, that we call representable, which classifies all semibiproducts of magmas whose behaviour is similar to semigroups, it is nevertheless more general than the subclass of associative magma-actions. [ABSTRACT FROM AUTHOR]

Enumerating the isomorphism or equivalence classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory. In 1988, Hofmeister enumerated the double covers of a graph, and this work was extended to n-fold coverings of a graph by Kwak and Lee. For regular graph coverings, Kwak, Chun and Lee enumerated the isomorphism classes of graph coverings when the covering transformation group is a finite abelian or a dihedral group in Kwak et al. (SIAM J Discrete Math 11:273–285, 1998). In 2018, the isomorphism classes of graph coverings are enumerated when the covering transformation groups are Z 2 -extensions of a cyclic group. As a continuation of this work, we enumerate the isomorphism classes of coverings of a graph when the covering transformation groups are Z p -extensions of a cyclic group for an odd prime integer p. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the projectively coresolved Gorenstein flat dimension of a group G over a commutative ring R and we prove that this dimension enjoys all the properties of the cohomological and the Gorenstein cohomological dimension. We also provide good estimations for the Gorenstein global dimension of RG in terms of this dimension and the Gorenstein global dimension of R. Moreover, we study special cases of groups, such as ▪-groups, and show that for such a group every Gorenstein projective RG -module is Gorenstein flat when the global dimension of R is finite. [ABSTRACT FROM AUTHOR]

The -simplex equation was introduced by Zamolodchikov as a generalization of the Yang–Baxter equation which becomes the -simplex equation in this terms. In the present article, we suggest general approaches to construction of solutions of the -simplex equation, describe certain types of solutions, and introduce an operation that allows us to construct, under certain conditions, a solution of the -simplex equation from solutions of the -simplex equation and -simplex equation. We consider the tropicalization of rational solutions and discuss its generalizations. We prove that a solution of the -simplex equation on can be constructed from solutions of this equation on and if is an extension of a group by a group . We also find solutions of the parametric Yang–Baxter equation on with parameters in . We introduce ternary algebras for studying the 3-simplex equation and present examples of such algebras that provide us with solutions of the 3-simplex equation. We find all elementary verbal solutions of the 3-simplex equation on a free group. [ABSTRACT FROM AUTHOR]

We study the following question: under what conditions extension of one residually nilpotent group by another residually nilpotent group is residually nilpotent? We prove some sufficient conditions under which this extension is residually nilpotent. Also, we study this question for semi-direct products and, in particular, for extensions of free group by infinite cyclic group: F n ⋊ φ ℤ. We find conditions under which this group is residually nilpotent, find conditions under which this group has long lower central series. In particular, we prove that for n = 2 the length of the lower central series of F n ⋊ φ ℤ is equal to 2, ω or ω 2 . [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]

Every topological group G has, up to isomorphism, a unique minimal G -flow that maps onto every minimal G -flow, the universal minimal flow $M(G).$ We show that if G has a compact normal subgroup K that acts freely on $M(G)$ and there exists a uniformly continuous cross-section from $G/K$ to $G,$ then the phase space of $M(G)$ is homeomorphic to the product of the phase space of $M(G/K)$ with K. Moreover, if either the left and right uniformities on G coincide or G is isomorphic to a semidirect product $G/K\ltimes K$ , we also recover the action, in the latter case extending a result of Kechris and Sokić. As an application, we show that the phase space of $M(G)$ for any totally disconnected locally compact Polish group G with a normal open compact subgroup is homeomorphic to a finite set, the Cantor set $2^{\mathbb {N}}$ , $M(\mathbb {Z})$ , or $M(\mathbb {Z})\times 2^{\mathbb {N}}.$ [ABSTRACT FROM AUTHOR]

In this paper, we show that residually-finite-by-weakly-sofic extensions are weakly sofic. More precisely, we show that if in an exact sequence of groups 1 → N → K ... G → 1 the group G is residually finite and N is weakly sofic, then K is weakly sofic. [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]

The article studies finite groups indecomposable to subdirect product of groups (subdirectly irreducible groups), commutator subgroups of which are cyclic subgroups. The article proves that extensions of a primary cyclic group by any subgroup of its automorphisms completely describe the structure of non-primary finite subdirectly irreducible groups with a cyclic commutator subgroup. The following theorem is the main result of this article: a finite non-primary group is subdirectly irreducible with a cyclic commutator subgroup if and only if for some prime number $p\geq 3$ it contains a non-trivial normal cyclic $p$-subgroup that coincides with its centralizer in the group. In addition, it is shown that the requirement of non-primality in the statement of the theorem is essential.

A non-relativistic analogue of the conformal group, namely, 15-parameter extended Galileo-Newton group, which includes space-time transfers and gravitational transformations, spatial rotations, boosts and scale transformations, is considered. Based on its 12-parameter subgroup (without rotations), the nonrelativistic n-body problem is formulated by taking into account the well-known effect of time dilation in a gravitational field. [ABSTRACT FROM AUTHOR]

This paper presents a special (six-dimensional) representation of Lie algebra of the nonrelativistic analog of the conformal group, 15-parameter extended Galilean–Newtonian group. In contrast to the conventional (four-dimensional) representation, the found set of infinitesimal operators can be obviously extended to the representation of the Lie algebra of the nonrelativistic analogue of the 16-parameters extended conformal group. In conclusion, the principal point of Klein's Erlangen concept ("Given a manifold and a transformation group acting on it") and the author's thesis "space-time is a metaphysical ghost that should be excluded from the scientific description of nature" are compared. [ABSTRACT FROM AUTHOR]

A quandle is an algebraic system with a binary operation satisfying three axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the category of quandles to the category of groups, we prove that these functors map extensions of groups to extensions of quandles. Finally, we construct some natural group homomorphisms from the second cohomology of a group to the second cohomology of its core and conjugation quandles. [ABSTRACT FROM AUTHOR]

We show that any extension of an Abelian group corresponds to a solution of the parametric Yang–Baxter equation. This statement is a generalization of the well-known construction of a braided set in terms of group structure to the case of group extensions. We also show that this construction in the case of a semidirect product is a specialization of a more general construction using principal bundles and that the case of vector bundles considered earlier is an infinitesimal version of the case of a solution coming from the principal bundle structure. [ABSTRACT FROM AUTHOR]

We describe the structure of a generalized near-group fusion category and present an example of this class of fusion categories which arises from the extension of a Fibonacci category. We then classify slightly degenerate generalized near-group fusion categories. We also prove a structure result for braided generalized Tambara-Yamagami fusion categories. [ABSTRACT FROM AUTHOR]

Providing agricultural advice at scale poses operational challenges. Technology may help if repeating content reinforces learning for recipients and thus improves adoption, but risks reducing efficacy given limited customization and human interaction. We tested videos shared with female farmers in India as a supplement to standard human-provided extension services promoting a climate-smart practice, System Rice Intensification. The average treatment effects are large but imprecise because of non-normally distributed outcomes, specifically fat right tails. Weighted quantile regressions show that the imprecision in estimating an average treatment effect comes from farmers with output or yields in the upper quantiles. Both quantile regressions of the 25% and 50% quantiles and a Bayesian hierarchical model (robust to several priors) reveal positive treatment effects, and two subtreatments, one that reinforces information on labor costs from adoption and a second that presents role models to motivate adoption, lead to even higher estimated treatment effects on output., (© 2024 The Author(s).)

We study a class of strictly weakly integral fusion categories IN,ζ, where N ≥ 1 is a natural number and ζ is a 2N th root of unity, that we call N-Ising fusion categories. An N-Ising fusion category has Frobenius-Perron dimension 2N+1 and is a graded extension of a pointed fusion category of rank 2 by the cyclic group of order Z2N. We show that every braided N-Ising fusion category is prime and also that there exists a slightly degenerate N-Ising braided fusion category for all N > 2. We also prove a structure result for braided extensions of a rank 2 pointed fusion category in terms of braided N-Ising fusion categories. [ABSTRACT FROM AUTHOR]

We study a class of strictly weakly integral fusion categories IN,ζ, where N ≥ 1 is a natural number and ζ is a 2N th root of unity, that we call N-Ising fusion categories. An N-Ising fusion category has Frobenius-Perron dimension 2N+1 and is a graded extension of a pointed fusion category of rank 2 by the cyclic group of order Z2N. We show that every braided N-Ising fusion category is prime and also that there exists a slightly degenerate N-Ising braided fusion category for all N > 2. We also prove a structure result for braided extensions of a rank 2 pointed fusion category in terms of braided N-Ising fusion categories. [ABSTRACT FROM AUTHOR]

In (Acta Math Hangar 65(2):149–164, 1994), Milnes raised the problem that whether the Ellis group of a distal group is itself distal. In this paper, we give a positive answer to Milnes problem for the case of group extensions of Abelian groups. We also investigate the Ellis groups of a well-known family of distal compact right topological groups, namely the groups E T ∞ , E T k and E T k - 1 × T , where E T is the group of all endomorphisms on the unit circle T . [ABSTRACT FROM AUTHOR]

We investigate the universal cover of a topological group that is not necessarily connected. Its existence as a topological group is governed by a Taylor cocycle, an obstruction in 3-cohomology. Alternatively, it always exists as a topological 2-group. The splitness of this 2-group is also governed by an obstruction in 3-cohomology, a Sinh cocycle. We give explicit formulas for both obstructions and show that they are equal. [ABSTRACT FROM AUTHOR]

In this paper, a collection of completeness type properties of topological spaces is introduced. They are determined by families of open covers. It is shown that some of the completeness type properties imply isocompactness, and that some of them are productive. This allows to specify new cases when closed pseudocompact subsets of topological spaces are compact. The completeness type properties are also considered below in connection with group extensions of topological groups. [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]

We connect the discrete logarithm problem over prime fields in the safe prime case to the logarithmic derivative. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study co-prolongationsof central extensions. We construct the obstruction theory forco-prolongations and classify the equivalence classes of these by kernels of ahomomorphisms between 2-dimensional cohomology groups of groups.

We first consider two groups, F 0 = { g ∈ C [ [ z ] ] | g ( 0 ) ≠ 0 } under multiplication and F 1 = z F 0 under composition, where C [ [ z ] ] is the ring of formal power series over the complex field. It is known that the Riordan group R is isomorphic to the semidirect product F 0 ⋊ F 1 . It may be viewed as a group extension of F 1 by F 0 . In this paper, the group of three-dimensional Riordan arrays is obtained from an extension of the group R by F 0 . This concept extends to the group of multi-dimensional Riordan arrays. As an application, we illustrate the use of the three-dimensional Riordan array in multiple combinatorial sums. [ABSTRACT FROM AUTHOR]

When the benefit of making a correct decision is sufficiently high, even a slight increase in the probability of making such a decision justifies an increase in the number of decision makers. Applying a standard uncertain dichotomous choice benchmark setting, this study focuses on the relative desirability of two alternatives: adding individuals with capabilities identical to the existing ones and adding identical individuals with mean-preserving capabilities that depend on the states of nature. Our main result establishes that when the group applies the simple majority rule, variability in the capabilities of the new decision makers under the two states of nature, which is commonly observed in various decision-making settings, is less desirable in terms of the probability of making the correct decision. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the $( alpha, gamma)$-prolongation of central extensions. We obtain the obstruction theory for $( alpha, gamma)$-prolongations and classify $( alpha, gamma)$-prolongations thanks to low-dimensional cohomology groups of groups.