# corollaries of basic theorem on ordered groups

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

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

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 $$ if and only if $$.

Proof: We shall first prove that $$ implies $$ 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 $$ and $$ to obtain $$.

As for the proof that $$ implies $$, we shall prove the contrapositive statement. Assume that $$ is false. By conclusion 1, it follows that either $x=y$ or $x>y$. If $x=y$, then ${x}^{n}={y}^{n}$ so, by conclusion 1 $$ is false. If $x>y$ then, by what we have already shown, ${x}^{n}>{y}^{n}$ so $$ 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 $$ or $x=y$ or $$. If $$ then, by the preceding corollary, $$, which is not possible. Likewise, if $$, then we would have $$, 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 |