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Can you give some references to find all Sylow subgroups of $GL_n(\mathbb{F}_q)$? I know that upper triangular matrices with diagonal's 1 is a Sylow $p$-subgroup where $q=p^n$. But how about the other cases? Thanks!

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  • $\begingroup$ If you understand what an irreducible representation is, and how the Sylow subgroups of the symmetric group look like, you can work it out yourself quite easily. $\endgroup$
    – j.p.
    Nov 13, 2014 at 17:37

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See, for example, A. J. Weir's original paper "Sylow $p$-subgroups of the general linear group over finite fields of characteristic $p$" (link here).

P.S.: There are some obsolete notation in that article, for example, $A \circ B$ (so called "complete product"), it's just the wreath product of $A$ and $B$ ($A \wr B$).

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