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Have you tried Chapter 17, section 17.1 from Springer's book on algebraic groups? I believe that is as down-to-earth as it can get, and it is certainly rather detailed.

If $k$ is a commutative field of characteristic $p>0$, then the map $$ \theta \colon \mathrm{GL}_p(k) \to \mathrm{SL}_p(k) \colon A = (a_{ij}) \mapsto (\det A)^{-1} (a_{ij}^p) $$ is a group homomorphi …

The map $T \colon \mathrm{PSL}_2(p) \to \operatorname{Sym}(\mathbb{F}_{2^n}) \colon f \mapsto T_f$ does not have its image in $\mathrm{GL}_n(2)$ for other Mersenne primes $p = 2^n - 1$, unlike the cas …

The key word here is "polarity". A polarity of a projective plane with point set $P$ and line set $L$ is a map $\pi$ from $P \cup L$ to itself mapping points to lines and lines to points, such that $\ …

For a reference, you could use Theorem I.5.1 from C. Chevalley, The Algebraic Theory of Spinors, Columbia University Press, New York, 1954. (This also appears in volume 2 of his collected works.) The …

This is indeed known, and can be found, for instance, in the book "Algebraic combinatorics. I. Association schemes" by Bannai and Ito (1984), Section II.2, Example 2.1 (p. 53).

Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO. Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with $\operat …

There is a forthcoming book "Descent in buildings" by Bernhard Mühlherr, Holger Petersson and Richard Weiss, which should appear soon (published by Princeton University Press). See http://press.prince …

The most natural way to view your set of vectors is in the setting of projective geometry; the vector space $K^n$ (where $K$ is now an arbitrary field, so $K = \mathbb{C}$ for you) can be seen as a pr …

I'm not sure whether my answer is conceptual in your sense, but here is a relatively short proof. First of all, your definition of $f$ suggests the notation $$s_p := \sum_{i=p}^n x_i.$$ Now consider t …

Tits' original lecture notes from 1974 (Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, 386, Springer-Verlag) can still serve as a very good introduction to the subject. …

It's indeed a real pity that so many of Tits's papers are quite difficult to get. Luckily, there are some very concrete ideas by F. Buekenhout, B. Muehlherr, J.-P. Tignol and H. Van Maldeghem, to publ …

Dieudonné's "La Géometrie des Groupes Classiques" might be what you're looking for; it has a whole chapter on automorphisms and isomorphisms of the classical groups.