# 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. If $x<1$, then, by conclusion 5, $1^{-1}, so $x<1. If $x=1$, the conclusion is trivial. If $1, then, by conclusion 5, $x^{-1}<1^{-1}$, so $x^{-1}<1.

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 if and only if $x^{n}.

Proof: We shall first prove that $x implies $x^{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 and $x^{n} to obtain $x^{n+1}.

As for the proof that $x^{n} implies $x, we shall prove the contrapositive statement. Assume that $x 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} is false. If $x>y$ then, by what we have already shown, $x^{n}>y^{n}$ so $x^{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 or $x=y$ or $y. If $x then, by the preceding corollary, $x^{n}, which is not possible. Likewise, if $y, then we would have $y^{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 CorollariesOfBasicTheoremOnOrderedGroups 2013-03-22 14:55:12 2013-03-22 14:55:12 rspuzio (6075) rspuzio (6075) 12 rspuzio (6075) Corollary msc 20F60 msc 06A05