We give a short and uniform proof of a special case of Tits’ Centre Conjecture using a theorem of J-P. Serre [8] and a result from [1]. We consider fixed point subcomplexes X of the building X = X(G) of a connected reductive algebraic group G, where H is a subgroup of G.

In these notes, we present some fundamental results concerning flag varieties and their Schubert varieties. By a flag variety, we mean a complex projective algebraic variety X, homogeneous under a complex linear algebraic group. The orbits of a Borel subgroup form a stratification of X into Schubert cells. These are isomorphic to affine spaces; their closures in X are the Schubert varieties, ge...

We prove that if X is a meromorphic subvariety of a commutative meromorphic group G and X has finite stabilizer, then X is contained in a translate of an algebraic subgroup of G. We give an application with a “diophantine” flavour.

In this paper we investigate the group of homeomorphisms of topological spaces. A subgroup K of the group S(X) of all permutations of a set X is called t-representable on X if there exists a topology T on X such that the group of homeomorphisms of (X, T ) = K. We determine the t-representability of groups generated by a permutation which is a product of disjoint cycles having equal lengths. AMS...

Let X = H\G be a homogeneous space defined over a number field k, where G is a connected linear k-group satisfying certain conditions, and H is a connected k-subgroup of G. For a finite set S of places of k we define a finite abelian group AS(X) which is the defect of weak approximation for X with respect to S. We describe AS(X) in terms of the Brauer group of X.

An algebraic construction for constant dimension subspace codes is called orbit code. It arises as the orbits under the action of a subgroup of the general linear group on subspaces in an ambient space. In particular orbit codes of a Singer subgroup of the general linear group has investigated recently. In this paper, we consider the normalizer of a Singer subgroup of the general linear group a...

In this paper, we study the notion of solvable $L$-subgroup of an $L$-group and provide its level subset characterization and this justifies the suitability of this extension. Throughout this work, we have used normality of an $L$-subgroup of an $L$-group in the sense of Wu rather than Liu.

A nilpotent Cantor action is a minimal equicontinuous $\Phi \colon \Gamma \times \mathfrak {X} \to {X}$ on space $\mathfrak {X}$, where $\Gamma$ contains finitely-generated subgroup $\Gamma _0 \subset \Gamma$ of finite index. In this note, we show that these actions are distinguished among general actions: any effective finitely generated group space, which continuously orbit equivalent to acti...

If H is a subgroup of compact group G, the probability that random element commutes with G denoted by Pr(H,G). Let 〈g〉 stand for monothetic generated an g∈G and let K be G. We prove Pr(〈x〉,G)>0 any x∈K if only has open normal T such K/CK(T) torsion. In particular, x∈G virtually central-by-torsion, is, there G/Z(T) also deduce number corollaries this result.

Infinitely many elliptic curves over $$\mathbf{Q }$$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect their 2. Let $$N_j(X)$$ denote the number with at least j pairs 4, and height most X. In this article we show that $$N_1(X) = c_{1,1}X^{1/3}+c_{1,2}X^{1/6}+O(X^{0.105})$$ . We also show, as $$X\rightarrow \infty $$ , $$N_2(X)=c_{2,1}X^{1/6}+o(X^{1/1...