{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $\\varphi : G \\to H$ be a group homomorphism, and let $K = \\ker \\varphi$ be its [[kernel|0Z424HK8CQ]].\n2. [[$\\varphi(e) = e$ and $\\varphi(b^{-1}) = \\varphi(b)^{-1}$|08FKQD7NJ3]]. Thus $e \\in K$, and $K$ is not empty.\n3. Let $a, b \\in K$. Then $\\varphi(ab^{-1}) = \\varphi(a)\\varphi(b)^{-1} = e e^{-1} = e$, so $ab^{-1} \\in K$. Thus $K$ is a subgroup of $G$.\n4. Let $k \\in K$ and $g \\in G$. Then $\\varphi(gkg^{-1}) = \\varphi(g)\\, e\\, \\varphi(g)^{-1} = e$.\n5. Thus $gKg^{-1} \\subseteq K$ for all $g \\in G$, and $K$ is a [[normal subgroup|0W77CVBCHQ]] of $G$.\n\n## Idea\n\nA homomorphism sends the identity to the identity and respects inverses. Thus the elements sent to the identity are closed under the group operations and under conjugation.\n",
  "commit": "f7cc1f60ee7d22cafedd3ef7b979a1abe7bcb828",
  "front": {
    "kind": "proof",
    "methods": [
      "0NTF6DMVAP"
    ],
    "msc": [
      "20A05"
    ],
    "name": "Proof that the kernel is a normal subgroup",
    "proves": "0C9ZKYSPYR",
    "sources": [
      {
        "locator": "Normal subgroups, quotient groups and homomorphisms",
        "url": "https://en.wikipedia.org/wiki/Normal_subgroup"
      }
    ],
    "tag": "AY60KQR8RT",
    "uses": [
      "0Z424HK8CQ",
      "0W77CVBCHQ",
      "08FKQD7NJ3"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/AY60KQR8RT/",
  "links": [
    {
      "field": "proves",
      "target": "0C9ZKYSPYR"
    },
    {
      "field": "uses",
      "target": "0Z424HK8CQ"
    },
    {
      "field": "uses",
      "target": "0W77CVBCHQ"
    },
    {
      "field": "uses",
      "target": "08FKQD7NJ3"
    },
    {
      "field": "methods",
      "target": "0NTF6DMVAP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/AY60KQR8RT.md",
  "msc": [
    "20A05"
  ],
  "name": "Proof that the kernel is a normal subgroup",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AY6/AY60KQR8RT.md",
  "status": null,
  "tag": "AY60KQR8RT",
  "uri": "m2://AY60KQR8RT",
  "url": "https://math2.tamnd.com/wiki/proof-that-the-kernel-is-a-normal-subgroup/",
  "writing": {
    "tag": "AY60KQR8RT",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}