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Is there a usable bound for the minimal index of a proper subgroup in a finite simple group of Lie type in terms of its rank and the characteristic (or even cardinality) of its field of definition?

One way to get such a bound would be to give a lower bound for the dimension of a non-trivial representation (over $\mathbb Q$, or over $\mathbb C$ — I don't know if these are the same for these groups). Possibly such a bound is obvious from a good knowledge of Deligne–Lusztig theory but this is something i would like to avoid if possible.

My naïve guess would be $p^r$ (where $r$ is the rank), from the example of $\mathrm{PSL}_{r+1}(\mathbb F_p)$ where there is a subgroup of index roughly $p^r$ (the maximal parabolic with Levi component $\mathrm{GL}_r$) and it seems hard (impossible?) to come up with subgroups of lower index. For $r=1, 2$ one can actually prove that $p^r$ is the right order using a lower bound on the dimension of a nontrivial representation (proven by elementary means), I don't know whether this generalises to higher ranks.

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    $\begingroup$ Table 4 of this paper -- arxiv.org/pdf/1301.5166.pdf -- gives what you need. This is the most correct version that I am aware of. (Earlier sources had the odd mistake.) $\endgroup$
    – Nick Gill
    Jun 30, 2021 at 11:25
  • $\begingroup$ This table is not 100% correct either, they list $^2F_4(2)$ as simple, whereas it has index 2 simple subgroup $^2F_4(2)'$ with a permutation representation of index 1600. $\endgroup$ Jun 30, 2021 at 13:33

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Explicit values for the minimum degree of a primitive permutation representation of a simple group of Lie type can be found in Table 4 of this paper:

Guest, Simon; Morris, Joy; Praeger, Cheryl E.; Spiga, Pablo, On the maximum orders of elements of finite almost simple groups and primitive permutation groups., Trans. Am. Math. Soc. 367, No. 11, 7665-7694 (2015). ZBL1330.20002.

The arXiv version is available here. As the authors explain, the entries in the table come from various sources, some of which were found to have the odd mistake. So far as I am aware no mistakes are known for the table as given.

One consequence of the values given in the table is that the degree of a primitive permutation representation of ${\rm PSL}_{r+1}(q)$ is bounded below by $q^r$ except when $(q,r)=(9,1)$.

Edit 18 March 2022: Apparently the entry for $E_7(q)$ in the paper above is wrong -- there is a $q^5-1$ in the formula which should be $q^5+1$. For the record the correct formula is given in this paper:

Vasil’ev, A. V., Minimal permutation representations of finite simple exceptional groups of types (E_6), (E_7), and (E_8), Algebra Logika 36, No. 5, 518-530 (1997); translation in Algebra Logic 36, No. 5, 302-310 (1997). ZBL0941.20006.

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  • $\begingroup$ they list $^2F_4(2)$ as simple in Table 4, isn't it a mistake? $\endgroup$ Jun 30, 2021 at 13:37
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    $\begingroup$ @DimaPasechnik, I think one should interpret the entries in the table as giving the min perm rep whenever the group listed in the first column is simple. In the subsequent proof they treat ${^2\!F_4}(2)'$ when they deal with the sporadic groups so, from that point of view, this group is not included in the table (and, indeed, the degree of a min perm rep of the simple group is smaller than for ${^2\!F_4}(2).$ In short, my take is that it is not a mistake but it certainly could be confusing. $\endgroup$
    – Nick Gill
    Jun 30, 2021 at 14:02
  • $\begingroup$ they do have $PSp_4(2)'$ in the table, so there is no consistency in this respect. $\endgroup$ Jun 30, 2021 at 16:29

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