= Archimedean linearly ordered groups are Abelian
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An <Archimedean ordered group> with a two-sided invariant <total order> is commutative. If there is a least positive element, its cofinal powers exhaust the group. Otherwise each $c>1$ has $d>1$ with $d^2<c$. To see this, choose $1<s<r<c$ and use $rs^{-1}$ if its square is below $r$; if not, the inequality $(rs^{-1})^2\geq r$ gives $s^2\leq r<c$. If positive $g,h$ satisfied $gh<hg$, put $c=(gh)^{-1}hg$ and choose such a $d$. Cofinality gives $d^m\leq g<d^{m+1}$ and $d^n\leq h<d^{n+1}$. Two-sided order invariance then gives both $hg<d^{m+n+2}$ and $hg=ghc>d^{m+n+2}$, a contradiction. Replacing negative elements by their inverses reduces arbitrary commutativity to the positive case.
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