corollaries of basic theorem on ordered groups

Corollary 1 Let $G$ be an ordered group. For all $x\in G$ , either $x\leq 1\leq x^{-1}$ or $x^{-1}\leq 1\leq x$ .

Proof: By conclusion 1, either $x<1$ or $x=1$ or $1<x$ . If $x<1$ , then, by conclusion 5, $1^{-1}<x^{-1}$ , so $x<1<x^{-1}$ . If $x=1$ , the conclusion is trivial. If $1<x$ , then, by conclusion 5, $x^{-1}<1^{-1}$ , so $x^{-1}<1<x$ .

Q.E.D.

Corollary 2 Let $G$ be an ordered group and $n$ a strictly positive integer. Then, for all $x,y\in G$ , we have $x<y$ if and only if $x^{n}<y^{n}$ .

Proof: We shall first prove that $x<y$ implies $x^{n}<y^{n}$ by induction. If $n=1$ , this is a simple tautology. Assume the conclusion is true for a certain value of $n$ . Then, conclusion 4 allows us to multiply the inequalities $x<y$ and $x^{n}<y^{n}$ to obtain $x^{n+1}<y^{n+1}$ .

As for the proof that $x^{n}<y^{n}$ implies $x<y$ , we shall prove the contrapositive statement. Assume that $x<y$ is false. By conclusion 1, it follows that either $x=y$ or $x>y$ . If $x=y$ , then $x^{n}=y^{n}$ so, by conclusion 1 $x^{n}<y^{n}$ is false. If $x>y$ then, by what we have already shown, $x^{n}>y^{n}$ so $x^{n}<y^{n}$ is also false in this case for the same reason.

Q.E.D.

Corollary 3 Let $G$ be an ordered group and $n$ a strictly positive integer. Then, for all $x,y\in G$ , we have $x=y$ if and only if $x^{n}=y^{n}$ .

Proof: It is trivial that, if $x=y$ , then $x^{n}=y^{n}$ . Assume that $x^{n}=y^{n}$ . By conclusion 1 of the main theorem, it is the case that either $x<y$ or $x=y$ or $y<x$ . If $x<y$ then, by the preceding corollary, $x^{n}<y^{n}$ , which is not possible. Likewise, if $y<x$ , then we would have $y^{n}<x^{n}$ , which is also impossible. The only remaining possibility is $x=y$ .

Q.E.D.

Corollary 4 An ordered group cannot contain any elements of finite order.

Let $x$ be an element of an ordered group distinct from the identity. By definition, if $x$ were of finite order, there would exist an integer such that $x^{n}=1$ . Since $1=1^{n}$ , we would have $x^{n}=1^{n}$ but, by Corollary 3, this would imply $x=1$ , which contradicts our hypothesis.

Q.E.D.

It is worth noting that, in the context of additive groups of rings, this result states that ordered rings have characteristic zero.

Title	corollaries of basic theorem on ordered groups
Canonical name	CorollariesOfBasicTheoremOnOrderedGroups
Date of creation	2013-03-22 14:55:12
Last modified on	2013-03-22 14:55:12
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	12
Author	rspuzio (6075)
Entry type	Corollary
Classification	msc 20F60
Classification	msc 06A05