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  "body": "\n## Summary\n\nAn orderable group, or bi-orderable group, is a [[group|G5S1M9]] that admits a total order preserved by multiplication on both sides.\nIn other words, it is a group that can be made into a [[linearly ordered group|P2E8BNYC7B]].\nOrderability is a property of the group, not extra structure, and a group can have many different orders.\nTorsion-free abelian groups and free groups are orderable, while groups with nontrivial elements of finite order are not.\n\n## Explanations\n\n### L1 human\n\nSome groups can have their elements lined up so that the line is respected by the group operation.\nThe integers and the rationals under addition are examples.\nA group with an element $g \\ne e$ and $g^n = e$ cannot be lined up like this.\nIf $g$ were larger than $e$, then $g^2$, $g^3$, and so on would be larger still, and $g^n = e$ could not happen.\n",
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