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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?

Rodrigo de Azevedo
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Thomas
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    If $|G:H|$ is finite then the Reidemeister-Schreier algorithm does this, and yes it is available in GAP. The general answer is no. – Derek Holt Jul 06 '20 at 07:21
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    It's also a strong 'no', in the sense that if $|G:H|$ is infinite then $H$ need not be finitely generated, even if $G$ is finitely presented. The group $\mathbb{Z}\wr \mathbb{Z}$ is such an example. – David A. Craven Jul 06 '20 at 15:20

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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, respectively IsomorphismFpGroupByGenerators

ahulpke
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