{
  "aliases": [
    "one-step-subgroup-test"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0ZTGMDNVTB"
    }
  ],
  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $H$ be a subset of $G$.\nThen $H$ is a [[subgroup|0X1JXT47GA]] of $G$ if and only if $H$ is not empty and $ab^{-1} \\in H$ for all $a, b \\in H$.\n\n## Remarks\n\nIf $a \\in H$, then $e = aa^{-1} \\in H$, then $b^{-1} = eb^{-1} \\in H$ for $b \\in H$, and then $ab = a(b^{-1})^{-1} \\in H$.\nIn additive notation, the condition is $a - b \\in H$ for all $a, b \\in H$.\nFor a finite nonempty subset $H$, it is sufficient that $ab \\in H$ for all $a, b \\in H$.\n\n## Explanations\n\n### L1 human\n\nThis test replaces three checks by one.\nYou only check that the subset is not empty and that $ab^{-1}$ stays in the subset.\nFor example, the multiples of $5$ form a subgroup of $(\\mathbb{Z}, +)$, because the difference of two multiples of $5$ is a multiple of $5$.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "aliases": [
      "one-step-subgroup-test"
    ],
    "kind": "statement",
    "msc": [
      "20A05"
    ],
    "name": "Subgroup criterion",
    "role": "proposition",
    "sources": [
      {
        "locator": "Subgroup tests",
        "url": "https://en.wikipedia.org/wiki/Subgroup"
      }
    ],
    "tag": "05N4TQKPH0",
    "uses": [
      "0N0PXCTYA7"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/05N4TQKPH0/",
  "links": [
    {
      "field": "uses",
      "target": "0N0PXCTYA7"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/05N4TQKPH0.md",
  "msc": [
    "20A05"
  ],
  "name": "Subgroup criterion",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/05N/05N4TQKPH0.md",
  "status": null,
  "tag": "05N4TQKPH0",
  "uri": "m2://05N4TQKPH0",
  "url": "https://math2.tamnd.com/wiki/subgroup-criterion/",
  "writing": {
    "tag": "05N4TQKPH0",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}