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  "body": "\n## Statement\n\nLet $(G, \\le)$ be a [[linearly ordered group|1P8087YGYQ]] with identity $e$.\nSuppose that $G$ is *Archimedean*: for all $a, b \\in G$ with $a \u003e e$, there is an integer $n \\ge 1$ with $b \u003c a^n$.\nThen there is an injective [[group homomorphism|0HV9Y7WG5R]] $\\varphi \\colon G \\to (\\R, +)$ such that $a \\le b$ if and only if $\\varphi(a) \\le \\varphi(b)$.\nIn particular, $G$ is abelian.\n\n## Remarks\n\nIf $G = \\set{e}$ there is nothing to prove.\nOtherwise one fixes an element $u \u003e e$ and sends $g$ to the real number $\\sup \\set{m/n : n \\ge 1, \\ u^m \\le g^n}$.\nThe Archimedean property makes this set bounded above and nonempty, and an argument with [[Dedekind cuts|04QNZZ4HHY]] shows that the map is an [[injective|0PVG1CRAJN]] order-preserving homomorphism.\nCommutativity of $G$ then follows from commutativity of $\\R$.\nExample: the subgroup $\\Z + \\Z\\sqrt{2}$ of $\\R$ is Archimedean and is already a subgroup of [[the reals|08J2M1JYAG]].\nThe Archimedean hypothesis is needed: $\\Z^2$ with the lexicographic order is linearly ordered, but $n \\cdot (0, 1) \u003c (1, 0)$ for all $n$.\nNo order-preserving injective homomorphism to $\\R$ exists for it, since the image of $(0, 1)$ would be positive with all its multiples below the image of $(1, 0)$.\n\n## Explanations\n\n### L1 human\n\nA linearly ordered group is Archimedean if no element is infinitely large compared with another positive element.\nHölder's theorem says that such a group is just a group of real numbers in disguise.\nSo the real numbers contain every way of measuring with a single scale and no infinitely large or small quantities.\n\n### L2 human\n\nThe theorem is the base case for the structure theory of linearly ordered abelian groups.\nA general linearly ordered abelian group splits into Archimedean classes, and the quotient attached to each class is Archimedean, so Hölder's theorem embeds it into $\\R$.\nThe [[Hahn embedding theorem|FVPXXFNVQY]] puts these real pieces together in a lexicographic product.\nThe theorem also explains why the value group of a rank one valuation can be taken inside $\\R$.\n",
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