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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $\\{e\\} = G_0 \\trianglelefteq \\cdots \\trianglelefteq G_n = G$ and $\\{e\\} = H_0 \\trianglelefteq \\cdots \\trianglelefteq H_m = G$ be two [[composition series|0XV2A49X6Q]] of $G$.\nThen $n = m$, and there is a permutation $\\sigma$ of $\\{1, \\dots, n\\}$ with $G_i / G_{i-1} \\cong H_{\\sigma(i)} / H_{\\sigma(i) - 1}$ for each $i$.\n\n## Remarks\n\nSo the [[composition factors|0PHTC93M58]] and the composition length of $G$ are well defined.\nFor $G = \\mathbb{Z}/n\\mathbb{Z}$, the theorem gives the uniqueness of the prime factorisation of $n$.\nOne proof uses the Schreier refinement theorem, which says that two subnormal series have equivalent refinements.\n\n## History\n\nThe theorem is named after Camille Jordan and Otto Hölder.\n\n## Explanations\n\n### L1 human\n\nA group can be cut into simple layers in many ways.\nThe theorem says that the layers that you get are always the same, up to order.\nThis is the group version of the uniqueness of prime factorisation.\n",
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