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  "body": "\n## Definition\n\nLet $\\langle S \\mid R \\rangle$ be a [[group presentation|0JGEQXQ4AC]], with $R$ a subset of the [[free group|0KTW26C80J]] $F(S)$.\nLet $N(R)$ be the [[normal closure|6XM4CCVTKE]] of $R$ in $F(S)$.\nThe *Tietze transformations* are the following four operations.\n\n1. Add a relation: replace $R$ by $R \\cup \\set{r}$, where $r \\in N(R)$.\n2. Remove a relation: replace $R$ by $R \\setminus \\set{r}$, where $r \\in R$ lies in the normal closure of $R \\setminus \\set{r}$.\n3. Add a generator: for a new symbol $x \\notin S$ and a word $w \\in F(S)$, replace $\\langle S \\mid R \\rangle$ by $\\langle S \\cup \\set{x} \\mid R \\cup \\set{x w^{-1}} \\rangle$.\n4. Remove a generator: suppose $x \\in S$ and $R$ contains a relation $x w^{-1}$, with $w$ a word in $S \\setminus \\set{x}$. Delete $x$ and this relation, and replace $x$ by $w$ in all other relations.\n\n## Remarks\n\nEach transformation yields a presentation of an [[isomorphic|08MG7H8JW5]] group.\nMoves 1 and 2 keep $N(R)$ unchanged, and in moves 3 and 4 the added or removed letter equals an existing element.\nTietze's theorem: two finite presentations define isomorphic groups if and only if one is obtained from the other by a finite sequence of Tietze transformations.\nExample: $\\langle a, b \\mid a b a^{-1} b^{-2},\\ b \\rangle$ becomes $\\langle a \\mid a a^{-1} \\rangle$ by removing $b$ with $w = 1$, and then $\\langle a \\mid \\ \\rangle$ by removing the trivial relation; so it presents $\\Z$.\nNon-example: replacing $\\langle a \\mid a^2 \\rangle$ by $\\langle a \\mid a^4 \\rangle$ is not a Tietze transformation, since $a^2$ is not in the normal closure of $a^4$; indeed the groups $\\Z/2$ and $\\Z/4$ differ.\nTietze transformations do not give an algorithm for the [[group isomorphism problem|MP3F49QT3C]], which is undecidable for finite presentations.\n",
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