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# 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.

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.

## Mathematics Subject Classification

20F60*no label found*06A05

*no label found*

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