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Let $G$ be a finite simple group and $p$ be a prime divisor of $|G|$. Let $V$ be a nontrivial irreducible $\mathbb{F}_p[G]$ module. I would like to understand the relation of $|V|$ and $|G|_p$ (the $p$-part of $|G|$). By checking the Atlas of Finite Groups, $|V|$ sometimes lesser than $|G|_p$ and sometimes larger than $|G|_p$. For example, if $|G|_p=p$ and $V$ is any nontrivial irreducible $\mathbb{F}_p[G]$-module, then $|V|>|G|_p$, if $G=\mathrm{SL}_4(2)$ and $V$ is a nontrivial irreducible $\mathbb{F}_2[G]$-module of dimension 4, then $|V|<|G|_2$.

Are there some inequalities to show the relation of $|V|$ and $|G|_p$?
For instance, could $|G|_p$ be bounded by a function of $|V|$? when $|V|$ divides $|G|$?

Any explanation, references suggestion and examples are appreciated.

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    $\begingroup$ Well $G \le {\rm GL}(V)$, so if $|V|=p^n$, then $|G|_p \le p^{n(n-1)/2}$, and the bound is sharp when $p=2$ and $G = {\rm GL}(V)$. $\endgroup$
    – Derek Holt
    Commented Mar 14, 2023 at 8:22
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    $\begingroup$ There's a related question of Richard Brauer, which asks whether the $p$-part of the dimension of an irreducible module in characteristic $p$ is at most the $p$-part of the group order. Thackray found a counterexample for the McLaughlin group, where there is a $2$-modular irreducible of dimension $7.2^9$, whereas the group order is only divisible by $2^7$. I later found an example for the symmetric group $S_{15}$, of dimension exactly $2^{12}$, whereas the group order is only divisible by $2^{11}$. $\endgroup$
    – Dave Benson
    Commented Mar 14, 2023 at 11:44
  • $\begingroup$ Thanks for your explanation @DerekHolt. $\endgroup$
    – user44312
    Commented Mar 16, 2023 at 1:22
  • $\begingroup$ Thanks for your explanation @DaveBenson. What do you mean by "2-modular irreducible of dimension"? Is that mean that the dimension of an irreducible $\mathbb{F}_2[G]$ module? $\endgroup$
    – user44312
    Commented Mar 16, 2023 at 1:23
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    $\begingroup$ I see. Thanks very much for your explanation. I have also found your paper. @Dave Benson "Benson, David. Some remarks on the decomposition numbers for the symmetric groups. Proc. Symp. Pure Math. Vol. 47. 1987." $\endgroup$
    – user44312
    Commented Mar 16, 2023 at 11:10

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