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If you have a finitely generated group given by a presentation, is there a good method to determine a presentation for a subgroup generated by some subset of the generators given in the presentation? Specifically: I have found a presentation for the group $GL_2(\mathbb{Z})$, which is $$\bigl\langle R,T,X\;\bigm|\; X^2=T^2=(XT)^4=(RT)^3=(RTX)^3=(RXT)^3=1\bigr\rangle,$$ with the matrices given by $$R=\left(\begin{array}{cc} 0 & 1\\1 & 1\end{array}\right),\space T=\left(\begin{array}{cc} 1 & 0\\0 & -1\end{array}\right),\space X=\left(\begin{array}{cc} 0 & 1\\1 & 0\end{array}\right)$$ Can one easily find a presentation for the subgroup generated by R and T?

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    Yes. Look up the Reidemeister-Schreier rewriting process, which can be found in Magnus, Karrass and Solitar "Combinatorial Group Theory". There is a geometric alternative, which is easier, but I forget its name. If your subgroup has finite index, then it too will be finitely presented (assuming the first group is). Otherwise, anything can happen (well, within reason...)!2011-08-23
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    I like to think of Reidemeister-Schreier as obtained by finding a 2-dimensional CW-complex whose fundamental group is the "big" group, then you find the covering space corresponding to the subgroup you're interested in, lift the CW-structure and compute its fundamental group.2011-08-24

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