scholarly journals A note on Dehn colorings and invariant factors

2018 ◽  
Vol 27 (14) ◽  
pp. 1871003 ◽  
Author(s):  
Derek A. Smith ◽  
Lorenzo Traldi ◽  
William Watkins
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Connected Sum ◽  
Classical Link ◽  
Invariant Factors ◽  
Torus Links

If [Formula: see text] is an abelian group and [Formula: see text] is an integer, let [Formula: see text] be the subgroup of [Formula: see text] consisting of elements [Formula: see text] such that [Formula: see text]. We prove that if [Formula: see text] is a diagram of a classical link [Formula: see text] and [Formula: see text] are the invariant factors of an adjusted Goeritz matrix of [Formula: see text], then the group [Formula: see text] of Dehn colorings of [Formula: see text] with values in [Formula: see text] is isomorphic to the direct product of [Formula: see text] and [Formula: see text]. It follows that the Dehn coloring groups of [Formula: see text] are isomorphic to those of a connected sum of torus links [Formula: see text].

scholarly journals Self-splitting Abelian groups

2001 ◽  
Vol 64 (1) ◽  
pp. 71-79 ◽  
Author(s):  
P. Schultz
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Free Module ◽  
Research Report ◽  
Original Form ◽  
Original Research ◽  
Direct Sum ◽  
The University ◽  
Torsion Free ◽  
Made In

G is reduced torsion-free A belian group such that for every direct sum ⊕G of copies of G, Ext(⊕G, ⊕G) = 0 if and only if G is a free module over a rank 1 ring. For every direct product ΠG of copies of G, Ext(ΠG,ΠG) = 0 if and only if G is cotorsion.This paper began as a Research Report of the Department of Mathematics of the University of Western Australia in 1988, and circulated among members of the Abelian group community. However, it was never submitted for publication. The results have been cited, widely, and since copies of the original research report are no longer available, the paper is presented here in its original form in Sections 1 to 5. In Section 6, I survey the progress that has been made in the topic since 1988.

scholarly journals Finite Groups Which Admit An Automorphism With Few Orbits

1960 ◽  
Vol 12 ◽  
pp. 73-100 ◽  
Author(s):  
Daniel Gorenstein
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Groups ◽  
Study Groups ◽  
Quaternion Group ◽  
Fixed Integer ◽  
Fixed Element ◽  
Odd Order ◽  
Special Case

In the course of investigating the structure of finite groups which have a representation in the form ABA, for suitable subgroups A and B, we have been forced to study groups G which admit an automorphism ϕ such that every element of G lies in at least one of the orbits under ϕ of the elements g, gϕr(g), gϕrϕ(g)ϕ2r(g), gϕr(g)ϕr2r(g)ϕ3r(g), etc., where g is a fixed element of G and r is a fixed integer.In a previous paper on ABA-groups written jointly with I. N. Herstein (4), we have treated the special case r = 0 (in which case every element of G can be expressed in the form ϕi(gj)), and have shown that if the orders of ϕ and g are relatively prime, then G is either Abelian or the direct product of an Abelian group of odd order and the quaternion group of order 8.

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

scholarly journals Certain Groups of Orthonormal Step Functions

1957 ◽  
Vol 9 ◽  
pp. 413-425 ◽  
Author(s):  
J. J. Price
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Compact Abelian Group ◽  
Walsh Functions ◽  
Cyclic Groups ◽  
Step Functions ◽  
Product Of Groups

It was first pointed out by Fine (2), that the Walsh functions are essentially the characters of a certain compact abelian group, namely the countable direct product of groups of order two. Later Chrestenson (1) considered characters of the direct product of cyclic groups of order α (α = 2, 3, …). In general, his results show that the analytic properties of these generalized Walsh functions are basically the same as those of the ordinary Walsh functions.

A characteristic property of elementary 2-groups of rank six

2011 ◽  
Vol 48 (3) ◽  
pp. 354-370
Author(s):  
Sándor Szabó
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Characteristic Property ◽  
Identity Element ◽  
Finite Abelian Group

Consider a finite abelian group G which is a direct product of its subsets A and B both containing the identity element e. If the non-periodicity of A and B forces that neither A nor B can span the whole G, then G must be an elementary 2-group of rank six.

scholarly journals Integral Cayley Graphs Defined by Greatest Common Divisors

10.37236/581 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Direct Product ◽  
Undirected Graph ◽  
Cayley Graphs ◽  
Cyclic Groups ◽  
Greatest Common Divisors ◽  
Gamma S

An undirected graph is called integral, if all of its eigenvalues are integers. Let $\Gamma =Z_{m_1}\otimes \ldots \otimes Z_{m_r}$ be an abelian group represented as the direct product of cyclic groups $Z_{m_i}$ of order $m_i$ such that all greatest common divisors $\gcd(m_i,m_j)\leq 2$ for $i\neq j$. We prove that a Cayley graph $Cay(\Gamma,S)$ over $\Gamma$ is integral, if and only if $S\subseteq \Gamma$ belongs to the the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. It is also shown that every $S\in B(\Gamma)$ can be characterized by greatest common divisors.

On the ARF Invariant of an Involution

1970 ◽  
Vol 22 (3) ◽  
pp. 519-524 ◽  
Author(s):  
Peter Orlik
Keyword(s):  
Abelian Group ◽  
Smooth Manifold ◽  
Finite Abelian Group ◽  
Homotopy Equivalent ◽  
Smooth Manifolds ◽  
Connected Sum ◽  
Group Operation ◽  
Free Involution ◽  
Arf Invariant

Let Σ4k+1 denote a smooth manifold homeomorphic to the (4k + 1)-sphere, S4k+1k ≧ 1, and T: Σ4k+1 → Σ4k+1 a differentiate free involution. Our aim in this note is to derive a connection between the differentiate structure on Σ4k+1 and the properties of the free involution T.To be more specific, recall [5] that the h-cobordism classes of smooth manifolds homeomorphic (or, what is the same, homotopy equivalent) to S4k+1, k ≧ 1, form a finite abelian group θ4k+1 with group operation connected sum. The elements are called homotopy spheres. Those homotopy spheres that bound parallelizable manifolds form a subg roup bP4k+2 ⊂ θ4k+1. It is proved in [5, Theorem 8.5] that bP4k+2 is either zero or cyclic of order 2. In the latter case the two distinct homotopy spheres are distinguished by the Arf invariant of the parallelizable manifolds they bound.

Quantum isometry groups of dual of finitely generated discrete groups and quantum groups

2017 ◽  
Vol 29 (03) ◽  
pp. 1750008 ◽  
Author(s):  
Debashish Goswami ◽  
Arnab Mandal
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Polynomial Growth ◽  
Finitely Generated ◽  
Isometry Groups ◽  
Product Decomposition ◽  
Full Spectrum ◽  
Spectral Triples ◽  
Growth Property ◽  
Quantum Isometry Groups

We study quantum isometry groups, denoted by [Formula: see text], of spectral triples on [Formula: see text] for a finitely generated discrete group [Formula: see text] coming from the word-length metric with respect to a symmetric generating set [Formula: see text]. We first prove a few general results about [Formula: see text] including: • For a group [Formula: see text] with polynomial growth property, the dual of [Formula: see text] has polynomial growth property provided the action of [Formula: see text] on [Formula: see text] has full spectrum. •[Formula: see text] for any discrete abelian group [Formula: see text], where [Formula: see text] is a suitable metric on the dual compact abelian group [Formula: see text]. We then carry out explicit computations of [Formula: see text] for several classes of examples including free and direct product of cyclic groups, Baumslag–Solitar group, Coxeter groups etc. In particular, we have computed quantum isometry groups of all finitely generated abelian groups which do not have factors of the form [Formula: see text] or [Formula: see text] for some [Formula: see text] in the direct product decomposition into cyclic subgroups.

scholarly journals A note on groups with non-central norm

1994 ◽  
Vol 36 (1) ◽  
pp. 37-43 ◽  
Author(s):  
R. A. Bryce ◽  
L. J. Rylands
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Quaternion Group ◽  
Order Four

The norm K(G) of a group G is the subgroup of elements of G which normalize every subgroup of G. Under the name kern this subgroup was introduced by Baer [1]. The norm is Dedekindian in the sense that all its subgroups are normal. A theorem of Dedekind [5] describes the structure of such groups completely: if not abelian they are the direct product of a quaternion group of order eight and an abelian group with no element of order four. Baer [2] proves that a 2-group with non-abelian norm is equal to its norm.

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