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I'm interested in finding all groups for which the group operation (and inverse map) may be implemented using finite field arithmetic.

I've done some searching and have come across "algebraic groups", but I don't know enough algebraic geometry to know if this is exactly what I'm after.

If it is, what is known about finding all algebraic groups over finite fields?

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    $\begingroup$ What do you mean by "implementable using finite field arithmetic"? Do you mean that the group is a subset of $F^N$ for some finite field $F$ and that the group operation is given by a polynomial? $\endgroup$
    – Denis Nardin
    Commented Jan 26, 2019 at 18:51
  • $\begingroup$ Yes, I think that would include everything I'm interested in. Though I think the group operation should be given by $N$ polynomials in $\mathbb{F}[x_1, \ldots, x_N]$ and likewise for the inverse map. $\endgroup$
    – user135066
    Commented Jan 26, 2019 at 19:00
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    $\begingroup$ @user135066 What you want is usally called "group of rational points in an algebraic group (over a finite field)". Another useful term is "groups of Lie type". I don't think they've been all classified. $\endgroup$
    – Denis Nardin
    Commented Jan 26, 2019 at 19:41
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    $\begingroup$ @DenisNardin I think that Suzuki and Ree groups are considered as groups of Lie type, and are not just defined as groups of points of an algebraic group over a finite field. It involves taking the fixed points by a group automorphism which is not induced by an algebraic automorphism. $\endgroup$
    – YCor
    Commented Jan 26, 2019 at 20:08
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    $\begingroup$ The question remains vague. First, it concerns finite groups if I guess correctly. Every finite group embeds into a symmetric group, which has an obvious linear representation over any field. So each group individually can somewhat be described using finite fields. Of course this is not very useful, but the question is not specific enough to emphasize what more you want to require. $\endgroup$
    – YCor
    Commented Jan 26, 2019 at 20:11

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