scholarly journals An embedding theorem for ordered groups

1975 ◽  
Vol 12 (3) ◽  
pp. 321-335 ◽  
Author(s):  
Colin D. Fox
Keyword(s):  
Positive Integer ◽  
Embedding Theorem ◽  
Normal Closure ◽  
Orderable Group ◽  
Ordered Groups

We show that if the normal, closure of an element a, of an orderable group, G, is abelian, then G can be embedded in an orderable group, G#, which contains an n-th root of a for every positive integer, n. Furthermore, every order of G extends to an order of G#.

Non-amalgamation of ordered groups

1984 ◽  
Vol 95 (2) ◽  
pp. 191-195 ◽  
Author(s):  
A. M. W. Glass ◽  
D. Saracino ◽  
C. Wood
Keyword(s):  
Positive Integer ◽  
Total Order ◽  
Ordered Group ◽  
Orderable Group ◽  
Ordered Groups

An ordered group (o-group for short) is a group endowed with a linear (i.e. total) order such that for all x, y, z, xz ≤ yz and zx ≤ zy whenever x ≤ y. A group for which such an order exists is called an orderable group. A group G is said to be divisible if for each positive integer m and each g ε G, there is x ε G such that xm = g.

scholarly journals On a problem in the theory of ordered groups

1972 ◽  
Vol 6 (3) ◽  
pp. 435-438 ◽  
Author(s):  
Colin D. Fox
Keyword(s):  
Cambridge Philos ◽  
Orderable Group ◽  
Counter Example ◽  
Ordered Groups ◽  
Defining Relation

The group G presented on two generators a, c with the single defining relation a−1c2a = c2a2c2 [proposed by B.H. Neumann in 1949 (unpublished), discussed by Gilbert Baumslag in Proc. Cambridge Philos. Soc. 55 (1959)] has been considered as a possible example of an orderable group which can not be embedded in a divisible orderable group, contrary to the conjecture that no such examples exist. It is known from Baumslag's discussion that G can not be embedded in any divisible orderable group. However, it is shown in this note that G is not orderable, and thus is not a counter-example to the conjecture.

scholarly journals Ordered groups, eigenvalues, knots, surgery and L-spaces

2011 ◽  
Vol 152 (1) ◽  
pp. 115-129 ◽  
Author(s):  
ADAM CLAY ◽  
DALE ROLFSEN
Keyword(s):  
Knot Theory ◽  
Real Root ◽  
Floer Homology ◽  
Heegaard Floer Homology ◽  
Necessary Condition ◽  
Finitely Generated ◽  
Positive Real ◽  
Orderable Group ◽  
Positive Real Root ◽  
Ordered Groups

AbstractWe establish a necessary condition that an automorphism of a nontrivial finitely generated bi-orderable group can preserve a bi-ordering: at least one of its eigenvalues, suitably defined, must be real and positive. Applications are given to knot theory, spaces which fibre over the circle and to the Heegaard–Floer homology of surgery manifolds. In particular, we show that if a nontrivial fibred knot has bi-orderable knot group, then its Alexander polynomial has a positive real root. This implies that many specific knot groups are not bi-orderable. We also show that if the group of a nontrivial knot is bi-orderable, surgery on the knot cannot produce an L-space, as defined by Ozsváth and Szabó.

An Embedding Theorem for Ordered Groups

1972 ◽  
Vol 24 (5) ◽  
pp. 944-946 ◽  
Author(s):  
Donald P. Minassian
Keyword(s):  
Partial Order ◽  
Direct Product ◽  
Embedding Theorem ◽  
Ordered Groups ◽  
Locally Nilpotent ◽  
Image Position ◽  
Torsion Free

An O*-group is a group wherein every partial order can be extended to some full order.THEOREM. Suppose the group G has a normal chain G = G1 ⊇ G2 ⊇ … such thatand each G/Gi is locally nilpotent and torsion-free. Then G can be embedded in thecomplete direct product G’ of divisible O*-groups.

scholarly journals An embedding theorem for ordered groups: Addendum

1977 ◽  
Vol 17 (3) ◽  
pp. 479-480
Author(s):  
Colin D. Fox
Keyword(s):  
Embedding Theorem ◽  
Simple Argument ◽  
Ordered Groups

A simple argument yields the following generalization of Theorem 6 of [1] (whose notation is retained without further explanation).

The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

On 4-Engel Groups

2007 ◽  
Vol 10 ◽  
pp. 341-353 ◽  
Author(s):  
Michael Vaughan-Lee
Keyword(s):  
Normal Closure ◽  
Engel Group

In this note, the author proves that a group G is a 4-Engel group if and only if the normal closure of every element g ∈ G is a 3-Engel group

Extensions of Rings Having McCoy Condition

2009 ◽  
Vol 52 (2) ◽  
pp. 267-272 ◽  
Author(s):  
Muhammet Tamer Koşan
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Nonzero Element ◽  
Ring Extensions

AbstractLet R be an associative ring with unity. Then R is said to be a right McCoy ring when the equation f (x)g(x) = 0 (over R[x]), where 0 ≠ f (x), g(x) ∈ R[x], implies that there exists a nonzero element c ∈ R such that f (x)c = 0. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if R is a right McCoy ring, then R[x]/(xn) is a right McCoy ring for any positive integer n ≥ 2.

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