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  "body": "\n## Description\n\nThe program in GAP lists each subgroup of $S_4$, the [[group|G5S1M9]] of the permutations of $\\{1, 2, 3, 4\\}$. It writes each subgroup with the sorted list of its elements to the certificate, and it prints the number of subgroups of each order.\n\nThe checker is a short Python program that does not use GAP. It finds the subgroups of $S_4$ again with its own search. It starts with the trivial subgroup and adds the subgroup that a known subgroup and one more element generate, until it finds no new subgroup. It accepts the certificate only when the lines of the certificate are these subgroups, each one time, and the order of each one divides 24.\n\n## Remarks\n\nThe orders divide 24, as [[Lagrange's theorem|3C53TD]] says. The converse of the theorem is not true for each finite group, but it is true for $S_4$. For each divisor $d$ of 24, $S_4$ has a subgroup of order $d$. The alternating group $A_4$ of order 12 has no subgroup of order 6, so the converse is false for $A_4$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "certificate": {
      "artefact": "blake3:2357b50189dc3ede995f7fd2f95cece3c9c18015d73b2388e9e69c37eb529b4f",
      "checker": "m2-python@sha256:84f284326896a049f22d31677fe9a7ee8dcc4d37261650c0a4b32e05f5e525f5",
      "checker_entry": "check.py",
      "kind": "witness-table"
    },
    "claim": "The symmetric group $S_4$ has 30 subgroups. It has 1, 9, 4, 7, 4, 3, 1 and 1 subgroups of the orders 1, 2, 3, 4, 6, 8, 12 and 24, and each of these orders divides 24.",
    "engine": {
      "entry": "compute/AAY/AAYEQE/main.g",
      "image": "m2-gap@sha256:1378718f9f1cbc151d7c44f33b246be3069087a55f8bfc44ea4fda4b0350d748",
      "resources": {
        "cpu": 1,
        "mem": "2G",
        "wall": "10m"
      },
      "seed": null
    },
    "kind": "computation",
    "name": "The subgroups of the symmetric group S4",
    "output": {
      "hash": "blake3:1f659e57b810fc209eeca3571bd7939cd6858d6a6eea06e77f448514ae842fd9",
      "summary": "subgroups of S4: 30. Orders 1, 2, 3, 4, 6, 8, 12 and 24: 1, 9, 4, 7, 4, 3, 1 and 1 subgroups."
    },
    "tag": "AAYEQE"
  },
  "kind": "computation",
  "link": "https://math2.tamnd.com/node/AAYEQE/",
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  "markdown": "https://math2.tamnd.com/node/AAYEQE.md",
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  "name": "The subgroups of the symmetric group S4",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/AAY/AAYEQE.md",
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  "tag": "AAYEQE",
  "uri": "m2://AAYEQE",
  "url": "https://math2.tamnd.com/wiki/the-subgroups-of-the-symmetric-group-s4/",
  "writing": {
    "tag": "AAYEQE",
    "kind": "computation",
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