Given a toric affine algebraic variety $X$ and collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ $\mathop{\rm Aut}(X)$ which are normalized by the torus acting on $X$, we show that group $G$ generated verifies following alternative Tits' type: either is group, or it contains non-abelian free subgroup. We deduce if $2$-transitive $G$-orbit in then subgroup, so, exponential growth.

We answer a question due to A. Myasnikov by proving that all expected ranks occur as the ranks of intersections of finitely generated subgroups of free groups. Mathematics Subject Classification (2000): 20E05 Let F be a free group. Let H and K be nontrivial finitely generated subgroups of F . It is a theorem of Howson [1] that H ∩K has finite rank. H. Neumann proved in [2] that rank(H ∩K)− 1 ≤ ...

OBJECTIVE To investigate the prevalence of subclinical hypothyroidism (SH) diagnosed by thyrotropin-releasing hormone (TRH) stimulating test in infertile women with basal thyroid-stimulating hormone (TSH) levels of 2.5 to 5.0 mIU/L. METHODS This study was performed in 39 infertile women with ovulatory disorders (group 1) and 27 infertile women with male infertility only (group 2, controls) wh...

Let $$p \geqslant 3$$ be a prime integer and, for $$l 1$$ , let $$G \cong {{\mathbb {Z}}}_{p}^{l}$$ group of conformal automorphisms some closed Riemann surface S genus $$g 2$$ . By the Riemann–Hurwitz formula, either \leqslant g+1$$ or $$p=2\,g+1$$ If $$l=1$$ and then S/G is sphere with exactly three cone points if moreover 11$$ G unique p-Sylow subgroup $$\textrm{Aut}\hspace{0.55542pt}(S)$$ $...

Let ‖ · ‖ be the euclidean norm on R and γn the (standard) Gaussian measure on R with density (2π)e 2/2. Let θ (≃ 1.3489795) be defined by γ1([−θ/2, θ/2]) = 1/2 and let L be a lattice in R n generated by vectors of norm ≤ θ. Then, for any closed convex set V in R with γn(V ) ≥ 1 2 and for any a ∈ R, (a + L) ∩ V 6= φ. The above statement can be viewed as a “nonsymmetric” version of Minkowski The...

The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always contains its own centraliser, so that any finite group is an abelian extension of a group of automorphisms of its generalised Fitting subgroup. We define a class of...

Elementarily free groups are the finitely generated groups with the same elementary theory as free groups. We prove that elementarily free groups are subgroup separable, answering a question of Zlil Sela. Limit groups arise naturally in the study of the set of homomorphisms to free groups and, in the guise of fully residually free groups, have long been studied in connection with the first-orde...

Let (G;F) be a connected reductive algebraic group over an algebraically closed field k of positive characteristic p, equipped with an Fq -structure coming from a Frobenius morphism F : G ! G. Let L : G ! G be the corresponding Lang map taking an element g 2 G to g 1F(g). By the Lang-Steinberg Theorem [Bor92, Theorem 16.3] this morphism of varieties is surjective with finite fibres. From this r...

A subgroup H of a group G is called ss-quasinormal in G if there is a subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B; H is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We fix in every non-cyclic Sylow subgr...

If H is a nitely generated subgroup of the nitely generated group G, then H is undistorted in G for their word metrics ii the induced homomorphism on the zero-dimensionaì 1-homology group is injective. The Mayer-Vietoris exact sequence for`1-homology is established and applications are given to hyperbolic groups and relative hyperbolicity. The notion of relative distortion of two maps is introd...