This is inspired by this question. Given finitely many invertible rational $n\times n$ matrices $A_{1},\ldots, A_{k}\in\operatorname{GL}(n,\mathbb{Q})$, is there an algorithm (a practical one) to determine whether the group $\langle A_{1},\ldots, A_{k}\rangle$ that they generate is finite? One could, I suppose, use something like Dimino's algorithm to calculate the closure and stop when the size exceeds the maximum order possible (which is the order $2^{n}n!$ of the group of signed permutation matrices, except for some small exceptions, if I recall correctly) but that seems impractical. Is there something better?
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There is such an algorithm, and it has allegedly been implemented in GAP and Magma. The only reference I have found is: László Babai, Robert Beals, and Daniel Rockmore. Deciding finiteness of matrix groups in deterministic polynomial time. In Proc. ISSAC'93 (Internat. Symp. on Symbolic and Algebraic Computation), Kiev 1993, pages 117-126. ACM Press, 1993. Unfortunately I could not find any open access version of this paper, and I have not seen it myself yet. |
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