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If $A$ is an arbitrary element of the general linear group $GL(2, q)$, is there a way to determine the order of $A$ without having to compute matrix powers?

The "brute force" method works fairly well. There are $q (q + 1) (q - 1)^2$ elements in $GL(2, q)$. At worst, you could check all divisors of this. Using this question it's possible to limit the search to the divisors of $q(q - 1)$ and $(q - 1)(q + 1)$. However, I am slightly unsatisfied at having to compute the order this way.

Robert D-B
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  • What don't you like about the answer already given for the question you ask? – amWhy Jun 11 '18 at 17:46
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    You'll need to provide more details about what you are actually looking for, if it even exists, else the cited duplicate already answers your question, despite your being "slightly unsatisfied at having to compute the order this way." – amWhy Jun 11 '18 at 17:48

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