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.



Subgroups and representations of finite groups of Lie type
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.**](http://dx.doi.org/10.1090/S0002-9947-2015-06293-X), Trans. Am. Math. Soc. 367, No. 11, 7665-7694 (2015). [ZBL1330.20002](https://zbmath.org/?q=an:1330.20002).


The arXiv version is available [here](https://arxiv.org/pdf/1301.5166.pdf). 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)**](https://eudml.org/doc/187826), Algebra Logika 36, No. 5, 518-530 (1997); translation in Algebra Logic 36, No. 5, 302-310 (1997). [ZBL0941.20006](https://zbmath.org/?q=an:0941.20006).


