corollaries of basic theorem on ordered groups

Corollary 1   Let G be an ordered group. For all xG, either x1x-1 or x-11x.

Proof:   By conclusionMathworldPlanetmath 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.


Corollary 2   Let G be an ordered group and n a strictly positive integer. Then, for all x,yG, we have x<y if and only if xn<yn.

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

As for the proof that xn<yn 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 xn=yn so, by conclusion 1 xn<yn is false. If x>y then, by what we have already shown, xn>yn so xn<yn is also false in this case for the same reason.


Corollary 3   Let G be an ordered group and n a strictly positive integer. Then, for all x,yG, we have x=y if and only if xn=yn.

Proof: It is trivial that, if x=y, then xn=yn. Assume that xn=yn. 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, xn<yn, which is not possible. Likewise, if y<x, then we would have yn<xn, which is also impossible. The only remaining possibility is x=y.


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

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


It is worth noting that, in the context of additive groupsMathworldPlanetmath 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