Extensions Of Abelian Groups Research Articles

Word automatic groups of nilpotency class 2

We consider word automaticity for groups that are nilpotent of class 2 and have exponent a prime p. We show that the infinitely generated free group in this variety is not word automatic. In contrast, the infinite extra-special p-group Ep is word automatic, as well as an intermediate group Hp which has an infinite centre. In the last section we introduce a method for showing automaticity of central extensions of abelian groups via co-cycles.

On the structure of symmetric n-ary bands

We study the class of symmetric [Formula: see text]-ary bands. These are [Formula: see text]-ary semigroups [Formula: see text] such that [Formula: see text] is invariant under the action of permutations and idempotent, i.e., satisfies [Formula: see text] for all [Formula: see text]. We first provide a structure theorem for these symmetric [Formula: see text]-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong [Formula: see text]-ary semilattice of [Formula: see text]-ary semigroups and we show that the symmetric [Formula: see text]-ary bands are exactly the strong [Formula: see text]-ary semilattices of [Formula: see text]-ary extensions of Abelian groups whose exponents divide [Formula: see text]. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric [Formula: see text]-ary band to be reducible to a semigroup.

On the embedding problem of central cyclic extensions of abelian groups

In this report we find the obstruction of the embedding problem related to a central cyclic extension of an arbitrary abelian group.

Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations

We consider elliptic operators associated with discrete groups of quantized canonical transformations. In order to be able to apply results from algebraic index theory, we define the localized algebraic index of the complete symbol of an elliptic operator. With the help of a calculus of semiclassical quantized canonical transformations, a version of Egorov's theorem and a theorem on trace asymptotics for semiclassical Fourier integral operators we show that the localized analytic index and the localized algebraic index coincide. As a corollary, we express the Fredholm index in terms of the algebraic index for a wide class of groups, in particular, for finite extensions of Abelian groups.

On the Ellis Group of a Class of Distal Groups and a Problem by Milnes

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 $$ E\left( {\mathbb{T}} \right)^{\infty } , E\left( {\mathbb{T}} \right)^{k} $$ and $$ E\left( {\mathbb{T}} \right)^{k - 1} \times {\mathbb{T}} $$, where $$ E\left( {\mathbb{T}} \right) $$ is the group of all endomorphisms on the unit circle $$ {\mathbb{T}} $$.

Convolution dominated operators on compact extensions of abelian groups

If $G$ is a locally compact group, $CD(G)$ the algebra of convolution dominated operators on $L^2(G)$, then an important question is: Is $\mathbb {C}1+CD(G)$ (or $CD(G)$ if $G$ is discrete) inverse-closed in the algebra of bounded operators on $L^2(G)$? In this note we answer this question in the affirmative, provided $G$ is such that one of the following properties is satisfied. There is a discrete, rigidly symmetric, and amenable subgroup $H\subset G$ and a (measurable) relatively compact neighbourhood of the identity $U$, invariant under conjugation by elements of $H$, such that $\{hU\;:\;h\in H\}$ is a partition of $G$.The commutator subgroup of $G$ is relatively compact. (If $G$ is connected, this just means that $G$ is an IN group.) All known examples where $CD(G)$ is inverse-closed in $B(L^2(G))$ are covered by this.

Generators of split extensions of Abelian groups by cyclic groups

Let G \simeq M \rtimes C be an n -generator group which is a split extension of an Abelian group M by a cyclic group C . We study the Nielsen equivalence classes and T-systems of generating n -tuples of G . The subgroup M can be turned into a finitely generated faithful module over a suitable quotient R of the integral group ring of C . When C is infinite, we show that the Nielsen equivalence classes of the generating n -tuples of G correspond bijectively to the orbits of unimodular rows in M^{n -1} under the action of a subgroup of GL _{n - 1}(R) . Making no assumption on the cardinality of C , we exhibit a complete invariant of Nielsen equivalence in the case M \simeq R . As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag–Solitar groups, split metacyclic groups and lamplighter groups.

On the classification of pointed fusion categories up to weak Morita equivalence

A pointed fusion category is a rigid tensor category with finitely many isomorphism classes of simple objects which moreover are invertible. Two tensor categories $C$ and $D$ are weakly Morita equivalent if there exists an indecomposable right module category $M$ over $C$ such that $Fun_C(M,M)$ and $D$ are tensor equivalent. We use the Lyndon-Hochschild-Serre spectral sequence associated to abelian group extensions to give necessary and sufficient conditions in terms of cohomology classes for two pointed fusion categories to be weakly Morita equivalent. This result may permit to classify the equivalence classes of pointed fusion categories of any given global dimension.

Endomorphisms in short exact sequences

ABSTRACTWe study the behaviour of endomorphisms and automorphisms of groups involved in abelian group extensions. The main result can be stated as follows: Let 0→N→G→Q→1 be an abelian group extension. Then one has the following exact sequence of groups:where EndN,Q(G) denotes the set of all endomorphisms of G which centralise N and induce identity on Q, denotes the set of all endomorphisms of G which normalise N and induce identity on Q and EndQ(N) denotes the set of endomorphisms of N which are compatible with the action of Q on N. This exact sequence is obtained using the five-term exact sequence in group cohomology.An interesting fact we discovered is that the first three terms involved have ring structure and the maps between them are ring homomorphisms. The ring structure on EndQ(N) is well-known, however the ring structure of the second term is a little more exotic. Restricted on quasi-regular elements, this gives the exact sequence proved recently in [4] by Passi, Singh and Yadav.

Multiplication formulas in Moufang loops

Using groups with triality, we obtain some general multiplication formulas in Moufang loops, construct Moufang extensions of abelian groups, and describe the structure of minimal extensions for finite simple Moufang loops over abelian groups.

Automorphisms of abelian group extensions

Let 1→N→G→H→1 be an abelian extension. The purpose of this paper is to study the problem of extending automorphisms of N and lifting automorphisms of H to certain automorphisms of G.

Ergodic properties of a class of discrete Abelian group extensions of rank-one transformations

We define a class of discrete Abelian group extensions of rank-one transformations and establish necessary and sufficient conditions for these extensions to be power weakly mixing. We show that all members of this class are multiply recurrent. We then stu

Axiomatic characterization of ordinary differential cohomology

Cheeger-Simons differential characters, Deligne cohomology in the smooth category, the Hopkins-Singer construction of ordinary differential cohomology and the recent Harvey-Lawson constructions are each in two distinct ways Abelian group extensions of known functors. In one desciption these objects are extensions of integral cohomology by the quotient space of all differential forms by the subspace of closed forms with integral periods. In the other they are extensions of closed differential forms with integral periods by the cohomology with coefficients in the circle. These two series of short exact sequences mesh with two interlocking long exact sequences (the Bockstein sequence and a DeRham sequence) to form a commutative DNA-like array of functors called the Character Diagram. Theorem 1.1 shows that on the category of smooth manifolds and smooth maps any package consisting of a functor into graded abelian groups together with four natural transformations that fit together so as to form a Character Diagram as above is unique up to unique natural equivalence. Theorem 1.2 shows the natural product structure on differential characters is uniquely characterized by its compatibility with the product structures on the known functors in the Character Diagram. The proof of Theorem 1.1 couples the naturality with results about approximating smooth singular cycles and homologies by embedded pseudomanifolds.

Description of the algebro-geometric Sturm–Liouville coefficients

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.

On the asymptotic behaviour of iterates of averages of unitary representations

Let $G$ be a locally compact group and $\mu$ a probability measure on $G$. Given a unitary representation $\pi$ of $G$, let $P_\mu$ denote the $\mu$-average $\int_G\pi(g)\,\mu(dg)$. $\mu$ is called neat if for every unitary representation $\pi$ and every $a$ in the support of $\mu$, $\slim_{n\to\infty}\bigl(P_\mu^n -\pi(a)^n E_\mu\bigr) =0$, where $E_\mu$ is a canonically defined orthogonal projection. $G\/$ is called neat if every almost aperiodic probability measure on $G$ is neat. Previously known results show that every almost aperiodic spread out probability measure is neat, in particular, every discrete group is neat; furthermore, identity excluding groups, in particular, compact groups and nilpotent groups, are neat. In this work neatness of solvable Lie groups, connected algebraic groups, Euclidian motion groups, [SIN] groups, and extensions of abelian groups by discrete groups is established. Neatness of ergodic probability measures on any locally compact group is also proven. The key to these results is the result that when $\{X_n\}_{n=1}^\infty$ is the left random walk of law $\mu$ on $G$ and $\pi$ a unitary representation in a separable Hilbert space, then for every $k=0,1,\dots$\,, the sequence $\pi(X_n)^{-1}P_\mu^{n-k}$ converges almost surely in the strong operator topology.

Extensions of symmetric cat-groups

This paper is an attempt to study extensions of symmetric categorical groups from a structural point of view. Using in a systematic way bilimits in the 2-category of symmetric categorical groups, we develop a theory which closely follows the classical theory of abelian group extensions. The basic results are established for any proper class of extensions, and a cohomological classification is obtained for those extensions whose epi part has a categorical section.

Commutative ideal extensions of Abelian groups

Commutative ideal extensions of Abelian groups

Commutative ideal extensions of abelian groups

Ideal extensions of semigroups were introduced by Clifford in [1] and since then they have been widely studied (see for instance [2]). Our aim here is to characterize commutative ideal extensions of Abelian groups. We show that they are those commutative semigroups with an idempotent Archimedean element, or equivalently, those commutative semigroups E such that E/R is an Abelian group, where R is the least congruence that makes E/R cancellative. These characterizations give rise to an algorithm for deciding from a presentation of a finitely generated commutative monoid whether it is an ideal extension of an Abelian group. Finally we present a procedure that enables us to compute the set of idempotents of a finitely generated commutative monoid. All semigroups appearing in this paper are commutative and for this reason in the sequel we sometimes omit the adjective commutative whenever we refer to a commutative semigroup (the same holds for monoids). The authors would like to thank the referee for his/her comments and suggestions.

On Preservation of Stability for Finite Extensions of Abelian Groups

AbstractWe characterize preservation of superstability and ω‐stability for finite extensions of abelian groups and reduce the general case to the case of p‐groups. In particular we study finite extensions of divisible abelian groups. We prove that superstable abelian‐by‐finite groups have only finitely many conjugacy classes of Sylow p‐subgroups.Mathematics Subject Classification: 03C60, 20C05.

Locally supersolvable extensions of Abelian groups

An extension E of an Abelian group A by a locally supersolvable group is studied under the assumption that A has a finite series of subgroups normal in E, each of whose factors satisfies one of the following conditions: 1) min-E; 2) max-E; 3) the factor has finite rank and is torsion-free. It is proved that if E induces in the factors of the series in A hypercyclic groups of automorphisms and A is the locally supersolvable coradical of E, then A is complemented in E and all complements are conjugate in E. An analog of this result for locally nilpotent extensions of Abelian groups is also noted.