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Homeproof of basic theorem about ordered groups

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# proof of basic theorem about ordered groups

# Property 1:

Consider $ab^{{-1}}\in G$. Since $G$ can be written as a pairwise disjoint union, exactly one of the following conditions must hold:

$ab^{{-1}}\in S\qquad ab^{{-1}}=1\qquad ab^{{-1}}\in S^{{-1}}$ |

By definition of the ordering relation, $a<b$ if the first condition holds. If the second condition holds, then $a=b$. If the third condition holds, then we must have $ab^{{-1}}=s^{{-1}}$ for some $s\in S$. Taking inverses, this means that $ba^{{-1}}=s$, so $b<a$, or equivalently $a>b$. Hence, one of the following three conditions must hold:

$a<b\qquad a=b\qquad b<a$ |

# Property 2:

The hypotheses can be rewritten as

$ab^{{-1}}\in S\qquad bc^{{-1}}\in S$ |

Multiplying, and remembering that $S$ is closed under multiplication,

$ac^{{-1}}=(ab^{{-1}})(bc^{{-1}})\in S.$ |

In other words, $a<c$.

# Property 3:

Suppose that $a<b$, so $ab^{{-1}}=s\in S$. Then

$s=ab^{{-1}}=a1b^{{-1}}=acc^{{-1}}b^{{-1}}=(ac)(bc)^{{-1}}$ |

so $ac<bc$.

By the defining property of $S$, we have $csc^{{-1}}\in S$. Also,

$csc^{{-1}}=cab^{{-1}}c^{{-1}}=(ca)(cb)^{{-1}},$ |

hence $(ca)(cb)^{{-1}}\in S$, so $ca<cb$

# Property 4:

# Property 5:

By the hypothesis, $ab^{{-1}}=s\in S$. By the defining property, $b^{{-1}}sb\in S$. Since $b^{{-1}}sb=b^{{-1}}a$, we have $b^{{-1}}a\in S$. In other words, $b^{{-1}}<a^{{-1}}$.

# Property 6:

By definition, $a<1$ means that $a1^{{-1}}\in S$. Since $1^{{-1}}=1$ and $a1=a$, this is equivalent to stating that $a\in S$.

## Mathematics Subject Classification

20F60*no label found*06A05

*no label found*

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