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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 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.

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 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]

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]

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.

In [1] the authors constructed a seven term exact sequence in the cohomology of a group extension 1 → N → G → Q → 1 with coefficients in a G -module M . However, they were unable to establish the precise link between the maps in that sequence and the corresponding maps arising from the spectral sequence associated to the group extension and the G -module M . In this paper, we show that there is a close connection between [1] and the two earlier papers [6] and [7] . In particular, we show that the results in [6] and [7] entail that the maps of [1] other than the obvious inflation and restriction maps do correspond to the corresponding ones arising from the spectral sequence. [ABSTRACT FROM AUTHOR]

Given a connected graph X and a group G of its automorphisms we first introduce an approach for constructing all pairwise nonequivalent connected solvable regular coverings ℘ : X ˜ → X (that is, with a solvable group of covering transformations CT ( ℘ ) ) along which G lifts, up to a prescribed order n of X ˜ . Next, given a connected solvable regular covering ℘ : X ˜ → X by means of voltages and a group G ≤ Aut ( X ) that lifts along ℘, we consider algorithms for testing whether the lifted group G ˜ is a split extension of CT ( ℘ ) . In computational group theory, methods for testing whether a given extension of permutation groups splits are known. However, in order to apply the existing algorithms, X ˜ together with CT ( ℘ ) and G ˜ need to be constructed in the first place, which is far from optimal. Recently, an algorithm avoiding such explicit constructions has been proposed by Malnič and Požar (submitted for publication) . We here provide additional details about this algorithm and investigate its performance compared to the one using explicit constructions. To this end, a concrete dataset of solvable regular covers of graphs has been generated by the algorithm mentioned in the first paragraph. [ABSTRACT FROM AUTHOR]

We characterise amenability of a countable group in terms of the spectral radius of the Perron-Frobenius operator associated to a group extension of a countable Markov shift and a Hölder continuous potential. This extends a result of Day for random walks and recent work of Stadlbauer for dynamical systems. Moreover, we show that if the potential satisfies a symmetry condition with respect to the group extension, then the logarithm of the spectral radius of the Perron-Frobenius operator is given by the Gurevi?c pressure of the potential. [ABSTRACT FROM AUTHOR]

Abstract: The extension of majorization (also called the rearrangement ordering), to more general groups than the symmetric (permutation) group, is referred to as G-majorization. There are strong results in the case that G is a reflection group and this paper builds on this theory in the direction of subgroups, normal subgroups, quotient groups and extensions. The implications for fundamental cones and order-preserving functions are studied. The main example considered is the hyperoctahedral group, which, acting on a vector in , permutes and changes the signs of components. [Copyright &y& Elsevier]

Abstract: The main results of this note extend a theorem of Kesten for symmetric random walks on discrete groups to group extensions of topological Markov chains. In contrast to the result in probability theory, there is a notable asymmetry in the assumptions on the base. That is, it turns out that, under very mild assumptions on the continuity and symmetry of the associated potential, amenability of the group implies that the Gurevič-pressures of the extension and the base coincide whereas the converse holds true if the potential is Hölder continuous and the topological Markov chain has big images and preimages. Finally, an application to periodic hyperbolic manifolds is given. [Copyright &y& Elsevier]

After a brief survey of the theory of group extensions and, in particular, of automorphisms of group extensions, we describe some recent reduction theorems for the inducibility problem for pairs of automorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a symmetric continuous cohomology of topological groups. This is obtained by topologizing a recent construction due to Staic [23], where a symmetric cohomology of abstract groups is constructed. We give a characterization of topological group extensions that correspond to elements of the second symmetric continuous cohomology. We also show that the symmetric continuous cohomology of a profinite group with coefficients in a discrete module is equal to the direct limit of the symmetric cohomology of finite groups. In the end, we also define symmetric smooth cohomology of Lie groups and prove similar results. [ABSTRACT FROM AUTHOR]

In this paper, we state the notion of morphisms in the category of abelian crossed modules and prove that this category is equivalent to the category of strict Picard categories and regular symmetric monoidal functors. The theory of obstructions for symmetric monoidal functors and symmetric cohomology groups are applied to show a treatment of the group extension problem of the type of an abelian crossed module. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we construct a seven-term exact sequence involving the cohomology groups of a group extension. Although the existence of such a sequence can be derived using spectral sequence arguments, there is little knowledge about some of the maps occurring in the sequence, limiting its usefulness. Here we present a construction using only very elementary tools, always related to the notion of conjugation in a group. This results in a complete and usable description of all the maps, which we describe both on cocycle level as on the level of the interpretations of low dimensional cohomology groups (e.g. group extensions). [Copyright &y& Elsevier]

The aim of this paper is to study the (α,γ)-prolongation of central extensions. We obtain the obstruction theory for (α,γ)-prolongations and classify (α,γ)-prolongations thanks to low-dimensional cohomology groups of groups. [ABSTRACT FROM AUTHOR]

We construct some new cohomology theories for topological groups and Lie groups and study some of its basic properties. For example, we introduce a cohomology theory based on measurable cochains which are continuous in a neighbourhood of the identity. We show that if G and A are locally compact and second countable, then the second cohomology group based on locally continuous measurable cochains as above parametrizes the collection of locally split extensions of G by A. [ABSTRACT FROM AUTHOR]

Abstract: Given a principal bundle (each being compact, connected and oriented) and a G-invariant metric on P which induces a volume form , we consider the group of all unimodular automorphisms of P, and determines its Euler equation à la Arnold. The resulting equations turn out to be (a particular case of) the Euler–Yang–Mills equations of an incompressible classical charged ideal fluid moving on B. It is also shown that the group is an extension of a certain volume preserving diffeomorphisms group of B by the gauge group of P. [Copyright &y& Elsevier]

Abstract: We obtain an explicit description of the Wells map for the automorphism group of a group extension in the full generality and investigate the dependency of this map on group extensions. Some applications are given. [ABSTRACT FROM AUTHOR]

Abstract: Using Barnes–Wall lattices and 1-cocycles on finite groups of monomial matrices, we give a procedure to construct tricosine spherical codes. This was inspired by a 14-dimensional code which Ballinger, Cohn, Giansiracusa and Morris discovered in studies of the universally optimal property. Their code has 64 vectors and cosines . We construct the Optimism Code, a 4-cosine spherical code with 256 unit vectors in 16-dimensions. The cosines are . Its automorphism group has shape . The Optimism Code contains a subcode related to the BCGM code. The Optimism Code implies existence of a nonlinear binary code with parameters , a Nordstrom–Robinson code, and gives a context for determining its automorphism group, which has form . [Copyright &y& Elsevier]

Abstract: It was discovered recently that there is a class of Sturm–Liouville operators whose coefficients are related to algebro-geometric data via a construction analogous to that carried out in the 1970s for the Schrödinger operator by Dubrovin, Matveev, and Novikov. In this paper, we study the “algebro-geometric” Sturm–Liouville coefficients in detail. Emphasis is placed on their recurrence properties. We make systematic use of the theory of abelian group extensions. [Copyright &y& Elsevier]

Abstract: The main purpose of the present paper is to give an explicit description of the Wells map of a given group extension to the case of automorphisms acting trivially on the quotient group. From this we obtain some of new necessary and sufficient conditions for an automorphism of a normal subgroup of a group to extend to the group itself, with trivial action on the quotient group. [Copyright &y& Elsevier]

In this note, we study the torsion of extensions of finitely generated abelian by elementary abelian groups. When the action is trivial (mod p), we make a specific choice of a 1-cochain for a vanishing multiple of the cohomology class defining the extension and use it to completely describe the torsion of central extensions. As an application, one gets that, under the assumption of trivial action on homology, Zpr may act freely on (S1)k if and only if r⩽k, providing an alternative proof of the main theorem in [Trans. Amer. Math. Soc. 352 (6) (2000) 2689–2700] for central extensions. [Copyright &y& Elsevier]

We extend (with some generalizations) Bronshtein's theorems about the disjointness of F-, PE-, V-extensions of minimal topological transformation groups. [ABSTRACT FROM AUTHOR]

Let $$N$$ and $$K$$ be groups and let $$G$$ be an extension of $$N$$ by $$K$$ . Given a property $$P$$ of group compactifications, one can ask whether there exist compactifications $$N^\prime $$ and $$K^\prime $$ of N and K such that the universal $$P$$ -compactification of G is canonically isomorphic to an extension of $$N^\prime $$ by $$K^\prime $$ . We prove a theorem which gives necessary and sufficient conditions for this to occur for general properties $$P$$ and then apply this result to the almost periodic and weakly almost periodic compactifications of G. [ABSTRACT FROM AUTHOR]