Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups

1999 ◽  
Vol 42 (3) ◽  
pp. 335-343 ◽  
Author(s):  
Goansu Kim ◽  
C. Y. Tang
Keyword(s):  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Residual Finiteness ◽  
Necessary And Sufficient Condition ◽  
Residual Properties ◽  
Hnn Extensions ◽  
Subgroup Separability ◽  
Necessary And Sufficient

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.

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

1996 ◽  
Vol 16 (1) ◽  
pp. 45-50 ◽  
Author(s):  
Fabio Fagnani
Keyword(s):  
Finite Group ◽  
Topological Entropy ◽  
Abelian Groups ◽  
Topological Classification ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Abelian Groups ◽  
A Finite Group ◽  
Necessary And Sufficient

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.

Extensions of Endomorphisms from the Higher Centres

1967 ◽  
Vol 19 ◽  
pp. 1250-1258 ◽  
Author(s):  
Franklin Haimo
Keyword(s):  
Exact Sequence ◽  
Abelian Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

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.

Edwards-Walsh resolutions of complexes and Abelian groups

2000 ◽  
Vol 62 (3) ◽  
pp. 407-416 ◽  
Author(s):  
Katsuya Yokoi
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Simplicial Complexes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

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).

scholarly journals Endomorphism rings of non-abelian groups

1970 ◽  
Vol 3 (3) ◽  
pp. 349-352 ◽  
Author(s):  
B. C. McQuarrie ◽  
J. J. Malone
Keyword(s):  
Arbitrary Element ◽  
Abelian Groups ◽  
Torsion Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Endomorphism Rings ◽  
Necessary And Sufficient

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.

EMBEDDING THEOREMS FOR GROUPS PRESENTED VIA PARTIAL AUTOMORPHISMS: A GENERALIZATION OF SEMIDIRECT PRODUCTS AND HNN EXTENSIONS

2002 ◽  
Vol 12 (01n02) ◽  
pp. 51-84 ◽  
Author(s):  
AKIHIRO YAMAMURA
Keyword(s):  
Inverse Semigroups ◽  
Embedding Problem ◽  
Sufficient Condition ◽  
Embedding Theorems ◽  
Necessary And Sufficient Condition ◽  
Semidirect Products ◽  
Geometric Methods ◽  
Hnn Extensions ◽  
Van Kampen Diagrams ◽  
Necessary And Sufficient

We examine a certain embedding problem for groups that have a presentation described by partial automorphisms. Semidirect products and HNN extensions have such a presentation. The embedding problem is closely related to systems of partial automorphisms, which are formalized by the concept of inverse semigroups. A necessary and sufficient condition for a group to be embedded in a certain sense is obtained by geometric methods using van Kampen diagrams.

scholarly journals GROUP RINGS OF COUNTABLE NON-ABELIAN LOCALLY FREE GROUPS ARE PRIMITIVE

2011 ◽  
Vol 21 (03) ◽  
pp. 409-431 ◽  
Author(s):  
TSUNEKAZU NISHINAKA
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Free Groups ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hnn Extensions ◽  
Necessary And Sufficient ◽  
Countably Infinite

We prove that every group ring of a non-abelian locally free group which is the union of an ascending sequence of free groups is primitive. In particular, every group ring of a countable non-abelian locally free group is primitive. In addition, by making use of the result, we give a necessary and sufficient condition for group rings of ascending HNN extensions of free groups to be primitive, which extends the main result in [Group rings of proper ascending HNN extensions of countably infinite free groups are primitive, J. Algebra317 (2007) 581–592] to the general cardinality case.

scholarly journals On the extension of orders in ordered modules

1970 ◽  
Vol 2 (1) ◽  
pp. 81-88 ◽  
Author(s):  
P. Ribenboim
Keyword(s):  
Abelian Groups ◽  
Independent Set ◽  
Total Order ◽  
Sufficient Condition ◽  
Necessary And 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.

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

2003 ◽  
Vol 55 (6) ◽  
pp. 1302-1338 ◽  
Author(s):  
Takeshi Katsura
Keyword(s):  
Abelian Groups ◽  
Sufficient Condition ◽  
Crossed Products ◽  
Necessary And Sufficient Condition ◽  
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.

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

1974 ◽  
Vol 10 (2) ◽  
pp. 161-167 ◽  
Author(s):  
C.M. Bang
Keyword(s):  
Abelian Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Ordered Sets ◽  
Infinite Matrices ◽  
Direct Limits ◽  
Necessary And Sufficient

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.

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