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单词 CorollariesOfBasicTheoremOnOrderedGroups
释义

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.

Q.E.D.

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.

Q.E.D.

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.

Q.E.D.

Corollary 4 An ordered group cannot contain any elements offinite 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 aninteger such that xn=1. Since 1=1n, we would have xn=1n but, by Corollary 3, this would imply x=1, which contradictsour hypothesisMathworldPlanetmath.

Q.E.D.

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

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更新时间:2025/5/4 10:24:17