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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] with a [[presentation|0JGEQXQ4AC]] $\\langle S \\mid R \\rangle$.\nLet $H$ and $K$ be [[subgroups|0X1JXT47GA]] of $G$, and let $\\alpha \\colon H \\to K$ be a group [[isomorphism|08MG7H8JW5]].\nLet $t$ be a new symbol not in $S$.\nThe *HNN extension* of $G$ relative to $\\alpha$ is the group\n$$G *_{\\alpha} = \\langle S, t \\mid R,\\ t h t^{-1} = \\alpha(h) \\text{ for all } h \\in H \\rangle,$$\nwhere each $h \\in H$ and $\\alpha(h) \\in K$ is written as a word in $S$.\nEquivalently, $G *_{\\alpha}$ is the quotient of the [[free product|TM7QS7P7X9]] $G * \\langle t \\rangle$, with $\\langle t \\rangle$ infinite cyclic, by the [[normal subgroup|0CB0DX5BVV]] generated by the elements $t h t^{-1} \\alpha(h)^{-1}$ for $h \\in H$.\nThe element $t$ is the *stable letter*, and $H$ and $K$ are the *associated subgroups*.\n\n## Remarks\n\nThe natural map from $G$ to $G *_{\\alpha}$ is injective, so $G$ is a subgroup of $G *_{\\alpha}$; this is a theorem of Higman, Neumann and Neumann.\nBritton's lemma describes when a word in $S$ and $t$ represents the identity.\n\nExample: let $G = \\langle a \\mid \\ \\rangle \\cong \\Z$, $H = \\langle a^m \\rangle$ and $K = \\langle a^n \\rangle$ with $m, n$ nonzero integers, and $\\alpha(a^m) = a^n$.\nThe map $\\alpha$ is an isomorphism, since both subgroups are infinite cyclic and $\\alpha$ sends a generator to a generator.\nThe HNN extension is $\\langle a, t \\mid t a^m t^{-1} = a^n \\rangle$, the [[Baumslag–Solitar group|429JJ5ZNBY]] $BS(m, n)$.\nFor $m = n = 1$ it is $\\langle a, t \\mid t a t^{-1} = a \\rangle \\cong \\Z^2$.\n\nNon-example: the rule $t h t^{-1} = \\varphi(h)$ for a homomorphism $\\varphi \\colon H \\to K$ that is not injective is not an HNN extension.\nIf $\\varphi(h) = 1$ for some $h \\ne 1$, then $h = t^{-1} \\varphi(h) t = 1$ in the quotient, so $G$ does not embed.\n",
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    "kind": "definition",
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    "name": "HNN extension",
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  "link": "https://math2.tamnd.com/node/TACX13XZST/",
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  "markdown": "https://math2.tamnd.com/node/TACX13XZST.md",
  "msc": [
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  "name": "HNN extension",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/TAC/TACX13XZST.md",
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  "url": "https://math2.tamnd.com/wiki/hnn-extension-definition/",
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