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Let $M \in H \leq GL(n, \mathbb{Z})$. Is there an algorithm that computes either matrix generators or even a group presentation for $C_H(M)$ given generators or a presentation of $H$? Also is $C_H(M)$ necessarily finitely generated if $H$ is?

I realize that you can view $C_H(M)$ as the set of integer solutions of the equations $MAM^{-1} = A$ where $A \in H$, but I wasn't sure how one can find generators.

Edit: We can assume that $H$ has decidable membership problem. In the context that I am working in, $H$ will be the centralizer of a previous matrix so membership is solvable by the above logic. I also have no idea if anything even exists in the case $H = GL(n, \mathbb{Z})$.

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    $\begingroup$ This looks hopeless without some restrictions on $H$. For example, the membership problem in $H$ could be undecidable. $\endgroup$
    – Derek Holt
    Commented Oct 10, 2013 at 17:26
  • $\begingroup$ Derek - I just edited the post in regards to your comment. $\endgroup$
    – user41149
    Commented Oct 10, 2013 at 17:31

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In the case of $GL(n, \mathbb{Z})$ itself this is addressed in the answer to this question. As for arbitrary subgroups, in some cases they are finitely generated. For reflection group, there is the paper of Daniel Alcock (Reflection centralizers in Coxeter groups), I doubt anything intelligent can be said in general.

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