We prove that any finitely generated solvable group of zero entropy contains a nilpotent subgroup of finite index. In particular, any finitely generated solvable group of exponential growth is of uniformly exponential growth.

Nakamura [N] introduced the G-Hilbert scheme for a finite subgroup G ⊂ SL(3, C), and conjectured that it is a crepant resolution of the quotient C3/G. He proved this for a diagonal Abelian group A by introducing an explicit algorithm that calculates A-HilbC3. This note calculates A-HilbC3 much more simply, in terms of fun with continued fractions plus regular tesselations by equilateral triangl...

Let π1(C) be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic p > 0 of countable cardinality. Let N be a normal (resp. characteristic) subgroup of π1(C). Under the hypothesis that the quotient π1(C)/N admits an infinitely generated Sylow p-subgroup, we prove that N is indeed isomorphic to a normal (resp. characteris...

Let (W,S) be a Coxeter system, and let X be a subset of S. The subgroup of W generated by X is denoted by WX and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of WX in W is the subgroup of w in W such that wWXw ∩WX has finite index in both WX and wWXw . The subgroup WX can be decomposed in the form ...

We consider recursive sequences over GF(q) (the Galois field with q elements) generated by an irreducible polynomialfof degree 12. All roots of f have the same order e (called the exponent off) in the multiplicative group GF(qn)* = GF(q”) (0). Every nonzero recursive sequence generated byf is periodic with period e, and there are (up to translation) (4% 1)/e nonzerof-sequences [l], [2]. If e = ...

Let U/Q be a unitary group of Q-rank one so that the real points U(R)?U(n,1). The U is only quasi-split over Q if and n=1,2. cohomology congruence subgroup closely related to theory automorphic forms. This relation best captured in so-called spaces H?(U,C), natural module under action U(Af). paper gives structural account U(Af)-module structure part which generated by residues or derivatives Ei...

In the work of Rostami et al., Bogomolov multiplier a Lie algebra L over field Ω is defined as particular factor subalgebra exterior product L∧L. If finite dimensional, we identify this object certain subgroup second cohomology group H2(L,Ω) by deriving Hopf-Type formula. As an application, affirmatively answer two questions posed Kunyavskiĭ regarding invariance under isoclinism algebras and ex...

Let G be a finite p-group which has exactly one maximal subgroup H such that |H| > p. Then we have d(G) = 2, p = 2, H is a four-group, G is abelian of order 8 and type (4, 2), G is of class 3 and the structure of G is completely determined. This solves the problem Nr. 1800 stated by Y. Berkovich in [3]. We consider here only finite p-groups and our notation is standard (see [1]). If G is a p-gr...