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Given a reductive group $G$ over a finite field and a parabolic subgroup $P$ , I wonder what are the orbits in $G$ under the adjoint action of $P$. This should be standard, but I can only find results …

I was wondering: Given a variety over a finite field, say the projective plane or sphere over $\mathbb{F}_q$. Then I can try to define parallel transport along (geodesic) curves. In particular, I can …

The structure of finite simple groups of Lie type of arbitrary rank can be described well via BN-pairs. BN-pairs basically generalize the Bruhat decomposition of matrices into monomial $N$ and triangu …

I have frequently seen results like: There are 4 isomorphism types of finite subgroups of $SL_2(\mathbb{Z})$, namely $\mathbb{Z}_2,\mathbb{Z}_3,\mathbb{Z}_4,\mathbb{Z}_6$. I wonder what is known of …

I think it is one of the wunderful beauties of the representation theory of finite groups of Lie type $G(\mathbb{F_{p^n}})$ such as $GL_2(\mathbb{F_p})$ mentioned above, that irreducible representati …

Let $G$ be a finite group, $k$ the field of complex numbers. Are there (cohomologically nontrivial) group 2-cocycles $\sigma\in Z^2(G,k^\times)$ such that for all $g,h\in G$: $$\sigma(g,h)=\si …

I cannot answer your question, but point to the right algebraic framework in my opinion: There is a well worked out classical (but somewhat underestimated) theory of inseparable Galois extensions. It …

If $g_i$ with $i=1\ldots n$ is a (minimal) set of generators of $G$ and $\chi_g(h)=:\langle g,h\rangle$ is considered as a scalar product $G\times G\rightarrow k^\times$ (see BS's answer about Pontrya …

Wei Zhou's answer already gives a very good example and I agree with the above comments. However for a "way out" - I want to give you two classes of obtacles, if you additionally remove the first, you …

Thanx to Humphreys and Koen for providing, that rank 1,2 is "geometrically unsatisfying" in the sense of Tits buildings and suggesting the notion of split BN-pairs as a tightning. But looking at the s …

One special case of groups, where one certainly gets rather quickly explicit and nontrivial expressions should be finite or affine Coxeter groups, that are finite/infinite and defined by involution ge …

Indeed the question is too vague for a precise answer, but nevertheless somehat natural ;-) I want to give some more details and clearifications to the "hierarchy", that has been broached by Carnahan …