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  "body": "\n## Definition\n\nLet $G$ be a group and let $\\le$ be a [[total order|0316AYH1SC]] on $G$.\nThe pair $(G, \\le)$ is a *linearly ordered group*, or *totally ordered group*, if for all $a, b, c \\in G$,\n$$a \\le b \\implies ca \\le cb \\text{ and } ac \\le bc.$$\nEquivalently, it is an [[ordered group|A49BATWQEH]] whose order is total.\n\n## Remarks\n\nExample: $\\Z^2$ under addition with the [[lexicographic order|0G77XFBV00]] is a linearly ordered group.\nIf $(a, b) \\le (c, d)$, then either $a \u003c c$, or $a = c$ and $b \\le d$, and both cases survive adding $(x, y)$ to both sides.\nNon-example: $\\Z^2$ with the componentwise order is an ordered group, but $(1, 0)$ and $(0, 1)$ are not comparable, so the order is not total.\nIn terms of the positive cone $P = \\set{g : e \\le g}$, the order is total if and only if $G = P \\cup P^{-1}$.\nA linearly ordered group has no elements of finite order other than $e$: if $g \u003e e$, then $g^n \u003e e$ for all $n \\ge 1$.\n",
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