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  "body": "\n## Summary\n\nTietze transformations are four elementary moves on a [[group presentation|0JGEQXQ4AC]] that do not change the group up to isomorphism.\nThey add or remove a redundant relation, or add or remove a redundant generator.\nTwo finite presentations define isomorphic groups if and only if one can pass from one to the other by finitely many such moves.\nThey are the basic tool to simplify presentations and to recognise when two presentations give the same group.\n\n## Explanations\n\n### L1 human\n\nA presentation describes a group by letters and rules, and many different presentations describe the same group.\nTietze moves are the safe edits.\nYou may add a rule that already follows from the others, or delete such a rule.\nYou may introduce a new letter as an abbreviation for a word, or remove a letter that is only an abbreviation.\nFor example, $\\langle a, b \\mid b = a^2 \\rangle$ becomes $\\langle a \\mid \\ \\rangle$ by removing the abbreviation $b$, so it is the infinite cyclic group.\n",
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