If I have a group $G$ with a known presentation, and a subgroup $H$ generated by known elements in $G$, is there an algorithm to determine the presentation of $H$ in terms of $G$? Is this doable in GAP?
2026-03-27 00:10:51.1774570251
Finding presentation of subgroup in GAP
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If the index is finite, the Reidemeister-Schreier algorithm (as mentioned by Derek Holt) will find a presentation of $H$ in terms of (a subset of the) Schreier generators for $H$. If you want a presentation of $H$ in your chosen generators, the (slightly different) Modified Todd-Coxeter algorithm does so. An excellent description of both algorithms can be found in the Handbook of Computational Group Theory by Holt, Eick, OBrien. Implementation in GAP can be obtained through the operations
IsomorphismFpGroup, respectivelyIsomorphismFpGroupByGenerators