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Algebraic groups are very rich objects. As such, a large bag of examples against which one can test his intuition can be very helpful in learning the general theory.

What are some good didactic (counter-)examples in the theory of algebraic groups?

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    $\begingroup$ Do most Homework exercises at ams.org/open-math-notes/omn-view-listing?listingId=110662 (and read Appendix A there), read various appendices in ams.org/open-math-notes/omn-view-listing?listingId=110663 (e.g., Appendices I, H, L, M, Q -- esp. Q.4, S, T -- esp. T.3), and in the book Pseudo-reductive Groups (2nd ed.) read all Examples in Chapter 1 as well as Example A.3.8, section A.6, the discussion immediately after the statement of Proposition A.5.9, Example A.8.3, Remark A.8.16, Example B.2.3 (contrast with tori), and Example B.2.9. $\endgroup$
    – nfdc23
    Commented Mar 29, 2017 at 0:17
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    $\begingroup$ $\mu_p$, for understanding non-smooth group schemes. Multiplicative groups of division algebras, for understanding non-quasisplit groups. Restrictions of scalars along non-separable extensions, for understanding pseudo-reductive groups. $\endgroup$
    – LSpice
    Commented Mar 29, 2017 at 0:46

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