Extensions of Finite Abelian Groups

Inverse property of non-associative abelian extensions

Our paper deals with the investigation of extensions of commutative groups by loops so that the quasigroups that result in the multiplication between cosets of the kernel subgroup are T-quasigroups. We limit our study to extensions in which the quasigroups determining the multiplication are linear functions without constant term, called linear abelian extensions. We characterize constructively such extensions with left-, right-, or inverse properties using a general construction according to an equivariant group action principle. We show that the obtained constructions can be simplified for or…

Partial cohomology of groups and extensions of semilattices of abelian groups

We extend the notion of a partial cohomology group $H^n(G,A)$ to the case of non-unital $A$ and find interpretations of $H^1(G,A)$ and $H^2(G,A)$ in the theory of extensions of semilattices of abelian groups by groups.

局部紧阿贝尔群的S-纯扩张

S-pure Extensions of Locally Compact Abelian Groups

本文在局部紧阿贝尔群范畴中引入并研究s-纯子群与s-纯扩张的概念及其性质。

Abelian Extensions of the Group of Diffeomorphisms of a Torus

In this paper we construct abelian extensions of the group of diffeomorphisms of a torus. We consider the jacobian map, which is a crossed homomorphism from the group of diffeomorphisms into a toroidal gauge group. A pull-back under this map of a central 2-cocycle on a gauge group turns out to be an abelian cocycle on the group of diffeomorphisms. We show that in the case of a circle, the Virasoro-Bott cocycle is a pull-back of the Heisenberg cocycle. We also give an abelian generalization of the Virasoro-Bott cocycle to the case of a manifold with a volume form.

Non-abelian extensions of infinite-dimensional Lie groups

In this paper we study non-abelian extensions of a Lie group $G$ modeled on a locally convex space by a Lie group $N$. The equivalence classes of such extension are grouped into those corresponding to a class of so-called smooth outer actions $S$ of $G$ on $N$. If $S$ is given, we show that the corresponding set $\Ext(G,N)_S$ of extension classes is a principal homogeneous space of the locally smooth cohomology group $H^2_{ss}(G,Z(N))_S$. To each $S$ a locally smooth obstruction class $χ(S)$ in a suitably defined cohomology group $H^3_{ss}(G,Z(N))_S$ is defined. It vanishes if and only if ther…

Abelian extensions of infinite-dimensional Lie groups

In the present paper we study abelian extensions of connected Lie groups $G$ modeled on locally convex spaces by smooth $G$-modules $A$. We parametrize the extension classes by a suitable cohomology group $H^2_s(G,A)$ defined by locally smooth cochains and construct an exact sequence that describes the difference between $H^2_s(G,A)$ and the corresponding continuous Lie algebra cohomology space $H^2_c(\g,\a)$. The obstructions for the integrability of a Lie algebra extensions to a Lie group extension are described in terms of period and flux homomorphisms. We also characterize the extensions w…

On t-extensions of abelian groups

Let G and C be two discrete abelian groups. It is known that Pext(C,G) is a subgroup of Ext(C,G). In this paper, we introduce the t-extensions of G by C. We will show that the set of all t-extensions of G by C is a subgroup of Ext(C,G) which contains Pext(C,G). Fulp and Griffith [4] determined the LCA groups G such that Ext(C,G)=0 for all compact connected groups C. Using the group of t-extensions, we determine the LCA groups G such that Pext(C,G)=0 for all compact connected groups C.

Linear extensions of orders invariant under abelian group actions

Let G be an abelian group acting on a set X, and suppose that no element of G has any finite orbit of size greater than one. We show that every partial order on X invariant under $G$ extends to a linear order on X also invariant under G. We then discuss extensions to linear preorders when the orbit condition is not met, and show that for any abelian group acting on a set X, there is a linear preorder <= on the powerset PX invariant under G and such that if A is a proper subset of B, then A<B (i.e., A<=B but not B<=A).

Linear extensions of partial orders on Abelian groups

Partially ordered groups, also known as po-groups, are groups with a compatible partial order. Results from M.I. Zajceva and H.-H. Teh are combined in order to provide a full characterisation of linear order extensions of a given order on a group. In contrast to Teh this approach provides a method to discuss linear orders of different abelian rank in a uniform manner. This will be achieved by modelling the linear orders using hyperplanes in a real vector spaces. Among some additional remarks a construction of an archimedian directed order is given for every torsion free abelian group.

Connection Cochain of Abelian Extensions and Connection $1$-Forms

In this paper, we consider the concept of connection cochain of central extensions introduced by Moriyoshi and apply it to the abelian case. We will show the relationship between connection cochain and connection $1$-form of a principal bundle whose structure group is abelian.

Pure extensions of locally compact abelian groups

In this paper, we study the group Pext(C,A) for locally compact abelian (LCA) groups A and C. Sufficient conditions are established for Pext(C,A) to coincide with the first Ulm subgroup of Ext(C,A). Some structural information on pure injectives in the category of LCA groups is obtained. Letting K denote the class of LCA groups which can be written as the topological direct sum of a compactly generated group and a discrete group, we determine the groups G in K which are pure injective in the category of LCA groups. Finally we describe those groups G in K such that every pure extension of G by …

Hopf-Galois structures of isomorphic type on a non-abelian characteristically simple extension

Let $L/K$ be a finite Galois extension whose Galois group $G$ is non-abelian and characteristically simple. Using tools from graph theory, we shall give a closed formula for the number of Hopf-Galois structures on $L/K$ with associated group isomorphic to $G$.

On Kahler extensions of abelian groups

We show that any Kahler extension of a finitely generated abelian group by a surface group of genus g at least 2 is virtually a product. Conversely, we prove that any homomorphism of an even rank, finitely generated abelian group into the genus g mapping class group with finite image gives rise to a Kahler extension. The main tools come from surface topology and known restrictions on Kahler groups.

k生成交换群扩展的多项式时间同构测试

Polynomial-time isomorphism test for $k$-generated extensions of abelian groups

本文提出了一种多项式时间算法，用于测试交换群的循环扩展和简单群扩展的同构性，推广了已有研究成果。

Galois module structure of $p^{\text{th}}$ power classes of abelian extensions of local fields

In this paper, we describe the Galois module structure of $J=\mathbf{K}^{\times}/\mathbf{K}^{\times p}$, where $\mathbf{K}$ is an extension of a local field $\mathbf{k}$ containing a primitive $p$-th root of unity: for instance, if $\mathbf{K}/\mathbf{k}$ is a $p$-elementary abelian extension, we prove that $J$ is a module of constant Jordan type, with stable Jordan type $[1]^2$, which, in a way, extends the result of J. Mináč and J. Swallow. Also, we take profit from our proof by computing some invariants, which were previously introduced by A. Adem, W. Gao, D. B. Karageuzian and J. Mináč onl…

作为可解扩张上Hopf-Galois结构的类型出现的非阿贝尔单群

Non-abelian simple groups which occur as the type of a Hopf-Galois structure on a solvable extension

本研究确定了可解扩张上作为Hopf-Galois结构类型的有限非阿贝尔单群，同时给出了乘法群可解的斜brace中作为加法群的这类单群的完整列表。

Subshifts and colorings on ascending HNN-extensions of finitely generated abelian groups

For an ascending HNN-extension $G*_ψ$ of a finitely generated abelian group $G$, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal{A}^{G*_ψ}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag-Solitar groups $\mathrm{BS}(1,N)$, $N\ge2$, for which our results imply that a $\mathrm{BS}(1,N)$-SFT which contains a configuration with period $a^{N^\ell}$, $\ell\ge 0$, must contain a strongly periodic configuration with monochromatic $\mathbb{Z}$-sections. Then we study proper $n$-c…

相对Rota–Baxter群的阿贝尔扩张的Wells型正合序列

A Wells-like exact sequence for abelian extensions of relative Rota--Baxter groups

本文研究相对Rota–Baxter群阿贝尔扩张的自同构可诱导性问题，通过证明Wells型正合序列建立导子群、自同构群与第二上同调之间的精确结构关系。

挠阿贝尔群的泛共扩张

Universal co-Extensions of torsion abelian groups

本文证明满足Ext-small的Ab3阿贝尔范畴具有Ab4条件当且仅当每个对象都是Ext-泛的，并刻画了挠阿贝尔群范畴中的co-Ext-泛对象为内射群与各p-分量有界的约化群的直和。