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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nThe group $G$ is *orderable*, or *bi-orderable*, if there is a [[total order|0316AYH1SC]] $\\le$ on $G$ such that for all $a, b, c \\in G$,\n$$a \\le b \\implies ca \\le cb \\text{ and } ac \\le bc.$$\nThat is, $(G, \\le)$ is a [[linearly ordered group|1P8087YGYQ]] for some order $\\le$.\n\n## Remarks\n\nExample: $\\Z^2$ under addition is orderable, since the lexicographic order is total and translation invariant.\nNon-example: the cyclic group $\\Z/2\\Z$ is not orderable.\nIf $g$ is its nonzero element and $0 \u003c g$, then adding $g$ gives $g \u003c g + g = 0$, a contradiction; the case $g \u003c 0$ is the same.\nMore generally, an orderable group is a [[torsion-free group|VQ4SSKS53D]].\nEvery torsion-free [[abelian group|071DDFVG4J]] is orderable.\nEvery [[free group|020Z5SEN27]] is orderable.\nA group is orderable if and only if it has a subset $P$ with $PP \\subseteq P$, $gPg^{-1} = P$ for each $g \\in G$, and $G = P \\sqcup \\set{e} \\sqcup P^{-1}$.\n",
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