# Questions tagged [ordered-groups]

An ordered group is a group with a (partial) order which the group operation preserves.

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### What is an isolated subgroup? [closed]

" As in the case of finite numbers, the infinitesimal numbers form an isolated subgroup  of $R_{inf}$ of $^*R$"page 152 What does this sentence mean? What is an isolated subgroup?
1answer
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### Generalization of order of the product of elements in an abelian group as follows.

Let $(G,*)$ be a group. Let $a_i \in G, |a_i|=n_i, 1 \le i \le m$. Suppose $\gcd(n_i,n_j)=1$ and $a_ia_j=a_ja_i$, for all $i$ and $j$. Let $x=a_1*a_2*\ldots*a_m$. Show that $|x| = n_1n_2\ldots n_m$. ...
2answers
24 views

### Let $G$ be a group with $|a_i|=n_i$ for all $1\le i \le m$. Show that $|x|=n_1n_2\ldots n_m$ where $x$ defined as follows

Let $(G,*)$ be a group. Let $a_i \in G, |a_i|=n_i, 1 \le i \le m$. Suppose $\gcd(n_i,n_j)=1$ and $a_ia_j=a_ja_i$, for all $i$ and $j$. Let $x=a_1*a_2*\ldots*a_m$. Show that $|x| = n_1n_2\ldots n_m$. ...
1answer
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### Let $(G,*)$ be a group and $a \in G$. Suppose that $|a|=n$ and $n=mk$ for some positive integers $m$ and $k$. What is $|a^k|$?

Let $(G,*)$ be a group and $a \in G$. Suppose that $|a|=n$ and $n=mk$ for some positive integers $m$ and $k$. What is $|a^k|$? attempt: Let $a \in G$ such that $|a|=n$. Then, \begin{equation*} a^n = a^...
1answer
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### Order of an element in a group problem as follows

Let $(G,*)$ be a group, $a \in G$, and $|a|=p$, where $p$ is a prime. Prove that $|a^k|=p$ for all $1 \le k \lt p$. Prove that for all $m \in \Bbb N$, either $a^m = e_G$ or $|a^m|=p$, where $e_G$ is ...
2answers
81 views

### Let $a$ and $n$ be integers with $n>0$. Show that the additive order of $a$ modulo $n$ is $\frac{n}{\gcd(a,n)}$.

Let $a$ and $n$ be integers with $n>0$. Show that the additive order of $a$ modulo $n$ is $\frac{n}{\gcd(a,n)}$. attempt: Let $a$ and $n$ be integers with $n>0$. Let $m=\frac{n}{\gcd(a,n)}$. ...
1answer
35 views

### Divisible totally ordered additive abelian groups [closed]

Let $(G,+,\leq)$ be a divisible totally ordered additive abelian group and $g_{1},g_{2}\in G$. If for every integer $n>1$, $g_{1}\geq (1-\frac{1}{n})g_{2}$, Can we have $g_{1}\geq g_{2}$? Thanks ...
1answer
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### Understanding totally ordered abelian groups

Let $(\Lambda, \leq ,+)$ be a totally ordered abelian group. Say such an abelian group is simple if it has no nontrivial quotients (the only quotients are $0$ and itself). One might wish to understand ...
0answers
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### Question about ordered group and its order ideal.

I am reading a book entitled An introduction to the classification of amenable C*-algebras. It reads, Definition 3.3.1 An ordered group $(G,G_+)$ is an abelian group $G$ with a distinguished ...
1answer
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### Ordered Groups: Left Multiplication vs Right Multiplication

Given that $G$ is a linearly ordered group (bi-ordered). I want to try and understand the difference between the “size” of left multiplication vs right multiplication (which I have written below using ...
0answers
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### Solving conjugation equations in bi-ordered groups

A bi-ordered group is a group $G$ equipped with a partial order $<$ with: $f<g\Longleftrightarrow h f i < h g i$ for all $f,g,h,i \in G$. In particular this is the case for groups of strictly ...
2answers
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2answers
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### Subgroups of the rationals is either generated by one element or infinitely generated

The following situation is given: Consider $(\mathbb{Q},+)$ as an ordered abelian group: $x,y\in \mathbb{Q}, \; x\le y\; :\Leftrightarrow y-x\ge 0$. Let $\mathbb{Q}_+=\{c\in\mathbb{Q}\mid c\ge 0\}.$ ...
1answer
104 views

### When does an abelian “linearly ordered group ” has the property that any non-empty subset of the set of “non-negative” elements has a “least” element

By a theorem of F.W. Levi , we know that an abelian group can be equipped with a linear order (https://en.wikipedia.org/wiki/Linearly_ordered_group) iff the group is torsion free . So let $(G,\le )$ ...
1answer
95 views

1answer
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### Existence of a sequence converging to $0$

Let $(G,+,\le)$ be a partially ordered group (with identity $0$) and suppose that for each positive $g \in G$ there exists $g^\prime \in G$ such that $0<g^\prime<g$. Is it true that there ...
1answer
132 views

### Is there any order isomorphic function between $\mathbb N \times \mathbb Z$ to $\mathbb Z \times \mathbb N$?

Is there any order isomorphic function between $\mathbb N \times \mathbb Z$ to $\mathbb Z \times \mathbb N$? (With lexicographical order) Thanks
1answer
39 views

### Ordered ring and mutiplicative ordinal

If $(E,<)$ is a linear order, let $s(E)$ denote the supremum of the set of ordinals which (order-)embed in $(E,<)$. $s(E)$ is also the set of ordinals which embed in $E$ with a non cofinal range....
1answer
80 views

### Probability to find at least one alphabetically ordered subset of K elements in a set of N elements

I would like to know how to calculate the probability of finding an alphabetically ordered subset of at least K elements in a set of N alphabetical elements. For example, for a set of N letters from ...
1answer
819 views

### Intervals in divisible ordered groups

Is it true that if $(G,+,0,<)$ is a divisible ordered abelian group with at least two elements, then for $a,b >0 \in G$, there is an injective order preserving map from $[0;a)$ to $[0;b)$? It ...
1answer
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### On ordered abelian groups containing $\mathbb{Z}$

Let $\Delta$ be an ordered abelian group containing $\mathbb{Z}$ as a subgroup of index $e$. I need to show that for any positive element $\delta \in \Delta$, we have $e\delta \geq 1$. I have no ...
1answer
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### Can the order on an ordered, cancellative monoid be extended to its Grothendieck group?

Suppose we have an ordered, cancellative monoid and we wish to apply the Grothendieck group construction to it. Can the total order be extended to the larger group? Example: consider the ordered ...
2answers
271 views

### Partial order relation on the group of integers

We know that the usual $\leq$ is a partial order relation on the group of integers $\mathbb Z$ and $\mathbb Z$ is a totally ordered with this partial order relation. Is there any other partially order ...
0answers
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### Equivalent condition of spliting exact sequence of partially ordered groups

A short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C\rightarrow 0$ of partially ordered group, where $\alpha : A\rightarrow B$ and $\beta: B\rightarrow C$ are order homomorphism is ...
1answer
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### An $RO$-group which is not $O$-group

I was thinking of some example for an Right ordered group ( $RO$-group) which is not an $O-$group (Ordered group) i.e. not left ordered. I guess looking in matrix groups will be fruitful but how to ...