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The question itself is natural, but it's fairly elementary and has a clearcut answer in the literature on finite simple groups including the series of books by Gorenstein-Lyons-Solomon (and for small …

Here are some supplementary comments, in community-wiki format: For either the original Chevalley groups or the twisted variants, the concrete, detailed treatment in Roger Carter's 1972 book here is …

It's useful to look at a short paper by Tits Sur le groupe des automorphismes de certains groupes de Coxeter, J. Algebra 113 (1988), no. 2, 346-357, available online here. While he doesn't treat arb …

Concerning your first question for the Suzuki groups in characteristic 2, it's helpful to go back to the original announcement by Suzuki: A new type of simple groups of finite order, Proc. Nat. Acad. …

As Stefan says, there is a clear counterexample for the simple groups of these two types, which can equally well be confirmed by looking at the relevant character tables in the Atlas of Finite Groups …

Questions of this type have been raised about various finite groups of Lie type at MathOverflow previously, for example here. As Nick Gill's comment indicates, the work of E. Vvodin is worth consu …

EDIT: This responds to Geoff's comment (and my carelessness), but also adds a couple of other remarks. Geoff's answer and the comments might be clarified a little as follows. You are looking at rat …

To attempt an answer, I'll replace your notation with my own. One method is to work inside a corresponding semisimple (or reductive) algebraic group $G$ over an algebraically closed field, relative to …

To amplify Geoff's brief comment, the most standard source for details about (most of) the ordinary characters of a Ree group of type $G_2$ (specified by an odd power of $3$ at least $27$ which we cal …

Yes, what does "generic" mean for a finite group? Geordie is correct that the Steinberg representation is far from being a typical Deligne-Lusztig character. In fact, its unique features make it …

Direct finite group computations of cohomology of the finite groups of Lie type tend to be very sparse. The case $p=2$ has special interest for topologists and does provide some explicit results. …

To expand a little more on the original question and Greg's solid answer, "larger than some sufficiently large constant" already strikes me as ambiguous. To start with a specified constant seems impo …

I believe the answer to the basic question here is that no one expects to find two such non-isomorphic groups. However, even with the classification of finite simple groups (probably) in hand, the …

My inclination at first is to be skeptical: Is there any numerical evidence?. The setting of the question is perhaps nonstandard, since for finite groups of Lie type the starting point for this kind …

You should look up an older article by Stephen Gagola, Jr., but read some of the arguments skeptically (as I did a long time ago when exploring this question in a graduate introduction to finite group …