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Let $G = GL(n, q)$ be the general linear group over the finite field $\mathbb{F}_q$ with $q$ elements, where $q$ is a power of a prime $p$. Let $m$ be a positive integer dividing $q-1$. Suppose $\mathcal{C}(G, C_m)$ is the set of $G$-conjugacy classes of the collection of subgroups of $G$ that are both completely reducible and isomorphic to the cyclic group $C_m$ of order $m$

Is there an explicit formula for the cardinality $|\mathcal{C}(G, C_m)|$ in terms of $n$, $m$, and $q$? For a fixed $n$, what is the asymptotic behavior of $|\mathcal{C}(G, C_m)|$ as $m$ tends to infinity among the divisors of $q-1$?

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    $\begingroup$ Rational canonical form seems relevant here. Also, all cyclic subgroups of order $m$ are automatically completely reducible under your hypotheses. $\endgroup$
    – Geoff Robinson
    Commented Sep 12 at 4:15
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    $\begingroup$ I'm puzzled by the phrase "as $m$ tends to infinity among the divisors of $q-1$". $\endgroup$
    – Dave Benson
    Commented Sep 12 at 8:00
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    $\begingroup$ A generator of such a subgroup is diagonalizable, and the number of conjugates of the subgroup that it generates will depend on the distribution of its multiset of eigenvalues, so I would not expect there to be an explicit formula. $\endgroup$
    – Derek Holt
    Commented Sep 12 at 8:20
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    $\begingroup$ I’m voting to close this question because the pattern of behaviour across several questions from the same account resembles past offenders $\endgroup$
    – Yemon Choi
    Commented Sep 16 at 21:32

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