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What is the relationship between pseudo-finite groups and real algebraic groups?

Could you provide an example of a pseudo-finite real algebraic group and of a non pseudo-finite one, if any?

Thank you.

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    $\begingroup$ Do you have any reasons to suspect the two might be related in any interesting way? For a trivial example, any finite (e.g. trivial) group is isomorphic to a real algebraic group and pseudofinite. On the other hand, I think most likely a "generic" real algebraic group is not pseudo-finite. $\endgroup$
    – tomasz
    Commented Sep 26, 2014 at 1:33
  • $\begingroup$ @tomasz In "Euler characteristic in semialgebraic and other o-minimal groups", Strzebonski proves that the (o-minimal) euler characteristic of a semialgebraic group plays the role of the cardinality of a finite group, so I was wondering how similar real algebraic groups are to finite groups and what first order property of finite groups fails for a generic real algebraic group. $\endgroup$
    – Alessandro
    Commented Sep 26, 2014 at 12:53

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