Extensions of Endomorphisms from the Higher Centres

If 0 → A → C → B → 0 is an exact sequence of abelian groups, if ƒ is a 2-cocyle for this extension, if α ∈ End A, and if β ∈ End B, then a necessary and sufficient condition that α extend to an endomorphism γ of C which induces β is that (M) αƒ and ƒβ be cohomologous ; see Montgomery (2). We shall extend this result to the case where 1 → A → G → B → 1 is an exact sequence of groups and A is abelian.

Some results on the classification of expansive automorphisms of compact abelian groups

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700008701 ◽

1996 ◽

Vol 16 (1) ◽

pp. 45-50 ◽

Cited By ~ 10

Author(s):

Fabio Fagnani

Keyword(s):

Finite Group ◽

Topological Entropy ◽

Abelian Groups ◽

Topological Classification ◽

Sufficient Condition ◽

Compact Abelian Groups ◽

A Finite Group ◽

AbstractIn this paper we study expansive automorphisms of compact 0-dimensional abelian groups. Our main result is the complete algebraic and topological classification of the transitive expansive automorpisms for which the maximal order of the elements isp2for a primep. This yields a classification of the transitive expansive automorphisms with topological entropy logp2. Finally, we prove a necessary and sufficient condition for an expansive automorphism to be conjugated, topologically and algebraically, to a shift over a finite group.

Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups

Canadian Mathematical Bulletin ◽

10.4153/cmb-1999-039-4 ◽

1999 ◽

Vol 42 (3) ◽

pp. 335-343 ◽

Cited By ~ 8

Author(s):

Goansu Kim ◽

C. Y. Tang

Keyword(s):

Cyclic Subgroup ◽

Abelian Groups ◽

Nilpotent Groups ◽

Sufficient Condition ◽

Residual Finiteness ◽

Residual Properties ◽

Hnn Extensions ◽

Subgroup Separability ◽

AbstractWe derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties ofHNN-extensions of nilpotent groups with cyclic associated subgroups.

Edwards-Walsh resolutions of complexes and Abelian groups

10.1017/s000497270001892x ◽

2000 ◽

Vol 62 (3) ◽

pp. 407-416 ◽

Cited By ~ 1

Author(s):

Katsuya Yokoi

Keyword(s):

Abelian Group ◽

Abelian Groups ◽

Simplicial Complexes ◽

Sufficient Condition ◽

We give a necessary and sufficient condition for the existence of an Edwards-Walsh resolution of a complex. Our theorem is an extension of Dydak-Walsh's theorem to all simplicial complexes of dimension ≥ n + 2. We also determine the structure of an Abelian group with the Edwards-Walsh condition, (which was introduced by Koyama and the author).

Endomorphism rings of non-abelian groups

10.1017/s0004972700046049 ◽

1970 ◽

Vol 3 (3) ◽

pp. 349-352 ◽

Cited By ~ 5

Author(s):

B. C. McQuarrie ◽

J. J. Malone

Keyword(s):

Arbitrary Element ◽

Abelian Groups ◽

Torsion Group ◽

Sufficient Condition ◽

Endomorphism Rings ◽

In 1942 F.W. Levi described those groups in which any two inner automorphic images of an arbitrary element commute. Until recently it was not known whether there existed non-abelian groups with the property that any two endomorphic images of an arbitrary element commute. Now, R. Faudree has given examples of finite p–groups having this property. In this paper we give a necessary and sufficient condition that a torsion group which contains no elements of order 2 has this property. The technique of proof involves looking at certain near rings of functions on the group. The inspiration for the theorem comes from the “halving” technique used by B.H. Neumann in 1940.

Extensions and automorphisms of Lie algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498817501626 ◽

2016 ◽

Vol 16 (09) ◽

pp. 1750162 ◽

Cited By ~ 5

Author(s):

Valeriy G. Bardakov ◽

Mahender Singh

Keyword(s):

Exact Sequence ◽

Lie Algebra ◽

Lie Algebras ◽

Short Exact Sequence ◽

Sufficient Condition ◽

Nilpotent Lie Algebra ◽

Lie Algebra Cohomology ◽

Cohomology Of Lie Algebras ◽

Let [Formula: see text] be a short exact sequence of Lie algebras over a field [Formula: see text], where [Formula: see text] is abelian. We show that the obstruction for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text] lies in the Lie algebra cohomology [Formula: see text]. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text], where [Formula: see text] is a free nilpotent Lie algebra of rank [Formula: see text] and step [Formula: see text].

On the extension of orders in ordered modules

10.1017/s0004972700041617 ◽

1970 ◽

Vol 2 (1) ◽

pp. 81-88 ◽

Cited By ~ 5

Author(s):

P. Ribenboim

Keyword(s):

Abelian Groups ◽

Independent Set ◽

Total Order ◽

Sufficient Condition ◽

Necessary And Sufficient ◽

Ordered Ring ◽

The Given

We introduce the notion of a positively independent set of elements in an ordered module. With this concept we determine a necessary and sufficient condition which insures that on a strictly ordered module over a strictly ordered ring there exists a strict total order refining the given order. This generalizes a previous result of Fuchs, concerning the case of ordered abelian groups.As an application, let R be a strictly ordered totally ordered ring and let M be the R-module of all mappings from a set I into R, with pointwise order; then this order on M may be refined to a strict total order.

The Ideal Structures of Crossed Products of Cuntz Algebras by Quasi-Free Actions of Abelian Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-2003-050-6 ◽

2003 ◽

Vol 55 (6) ◽

pp. 1302-1338 ◽

Cited By ~ 6

Author(s):

Takeshi Katsura

Keyword(s):

Abelian Groups ◽

Sufficient Condition ◽

Crossed Products ◽

Cuntz Algebras ◽

Necessary And Sufficient ◽

Free Actions ◽

The Ideal

AbstractWe completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A. Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra andK-groups of our algebras.

Partially well-ordered sets of infinite matrices and closed classes of abelian groups

10.1017/s0004972700040806 ◽

1974 ◽

Vol 10 (2) ◽

pp. 161-167 ◽

Cited By ~ 1

Author(s):

C.M. Bang

Keyword(s):

Abelian Groups ◽

Sufficient Condition ◽

Ordered Sets ◽

Infinite Matrices ◽

Direct Limits ◽

We give a necessary and sufficient condition for a class of row-decreasing infinite matrices to be partially well-ordered with regard to the component-wise ordering. Then, using these matrices, we determine all the classes of abelian groups, closed under taking subgroups, direct limits, and isomorphic groups.