Given a Chevalley group ${\mathbf G}(q)$ and parabolic subgroup $P\subset {\mathbf G}(q)$, we prove that for any set $A$ there is certain growth of relatively to $P$, namely, either $AP$ or $PA$ much larger than $A$. Also, study question about the intersection $A^n$ with subgroups $P$ large $n$. We apply our method obtain some results on modular form Zaremba’s conjecture from theory continued f...

We give an efficient algorithm for Lang’s Theorem in split connected reductive groups defined over finite fields of characteristic greater than 3. This algorithm can be used to construct many important structures in finite groups of Lie type. We use an algorithm for computing a Chevalley basis for a split reductive Lie algebra, which is of independent interest.

We generalize Mai cev's correspondence [5] to other nilpotent groups, namely to certain maximal unipotent subgroups of Chevalley groups. If -2* is a root system and K is an infinite field of characteristic different to 2 or 3, then we show that a group G elementarily equivalent to U^,(K) is isomorphic to U_^(F), where F is a field elementarily equivalent to K .

We generalize Chevalley’s theorem about restriction Res : C[g]g → C[h]W to the case when a semisimple Lie algebra g is replaced by a quantum group and the target space C of the polynomial maps is replaced by a finite dimensional representation V of this quantum group. We prove that the restriction map Res : (Oq(G) ⊗ V )Uq(g) → O(H) ⊗ V is injective and describe the image.