Minimal Length Elements of Thompson's Group F

Elements of the group are represented by pairs of binary trees and the structure of the trees gives insight into the properties of the elements of the group. The review section presents this representation and reviews the known relationship between elements of F and binary trees. In the main section we give a method of determining the minimal lengths of elements of Thompson's group F in the two generator presentation

Thompson＇s Group F and the Linear Group GL∞（Z）

The authors study the finite decomposition complexity of metric spaces of H, equipped with different metrics, where H is a subgroup of the linear group GL_∞(ℤ). It is proved that there is an injective Lipschitz map φ: (F, d _S ) → (H, d), where F is the Thompson’s group, dS the word-metric of F with respect to the finite generating set S and d a metric of H. But it is not a proper map. Meanwhile, it is proved that φ: (F, d _S ) → (H, d ₁) is not a Lipschitz map, where d ₁ is another metric of H.     

On the Validity of Thompson's Conjecture for Finite Simple Groups

In this article, we prove a conjecture of J. G. Thompson for the finite simple group ² D _n (q). More precisely, we show that every finite group G with the property Z(G) = 1 and N(G) = N(² D _n (q)) is necessarily isomorphic to ² D _n (q). Note that N(G) is the set of lengths of conjugacy classes of G.     

On the Unit Groups of Commutative Modular Group Algebras of KT-Groups

ABSTRACT

In this article, we prove that the inner projection of a projective curve with higher linear syzygies has also higher linear syzygies. Specifically, if a very ample line bundle ? on a smooth projective curve X satisfies property N _p for p  ≥  1 and H ¹ (? ^? 2) =  0 , then ?( ?  q ) satisfies property N _{p ? 1} for any point q  ∈  X . We also give simple proofs of well-known theorems about syzygies and raise some questions related to the line bundles of degree 2 g which do not satisfy property N ₁ .     

Hypercentral Unit Groups and the Hyperbolicity of a Modular Group Algebra

We classify groups G such that the unit group 𝒰 ₁(? G) is hypercentral. In the second part, we classify groups G whose modular group algebra has hyperbolic unit groups 𝒰 ₁(KG).     

Free Groups and Subgroups of Finite Index in the Unit Group of an Integral Group Ring

In this article we construct free groups and subgroups of finite index in the unit group of the integral group ring of a finite non-Abelian group G for which every nonlinear irreducible complex representation is of degree 2 and with commutator subgroup G′ a central elementary Abelian 2-group.     

On the Dynamics of Homeomorphisms on the Unit Ball of Rn

Positivity - Consider a compact convex subset X of R n (n ≥ 2) with non-empty interior and let H(X) be the set of all homeomorphisms from X onto X endowed with the supremum metric. We are...     

Group Rings with Solvable Unit Groups of Minimal Derived Length

Let F be a field of characteristic p > 2 and G a nonabelian nilpotent group containing elements of order p. Write F G for the group ring. The conditions under which the unit group ??(F G) is solvable are known, but only a few results have been proved concerning its derived length. It has been established that if G is torsion, the minimum derived length is ?log₂(p + 1)?, and this minimum occurs if and only if |G′| = p. In the present note, we show that the same holds if G has elements of infinite order.     

Noetherian Skew Group Rings of F.C Groups

Let R *θG be the skew group ring with a F.C group G and the group homom-rphismθfrom G to Aut(R), the group of automorphisms of the ring R. In this paper,the necessary and sufficient condition such that R*θG will be Noetherian is given, which generalizes the results of I.G. connel.     

On Linear Groups of Degree 2n Containing a Representation of the Special Linear Group of Degree n

Let n be an integer, n ≥ 2, and let a field P be a quadratic extension of an infinite field k. Regarding P as a k-vector space of dimension 2, we consider an n-dimensional P-vector space V as a 2n-dimensional k-vector space so the general linear group GL _n (P) acting on V is embedded in the group GL _2n (k). Let a field K be an algebraic extension of k. In this article, we determine overgroups of the special linear group SL _n (P) in the group GL _2n (K).     

On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line

In [1] G. Margulis proved Ghys's conjecture stating the validity of the following analog of the Tits alternative: either the group of homeomorphisms of the circle possesses a free subgroup with two generators or there is an invariant probabilistic measure on S ¹ . In the present paper, we prove the following strengthening of Margulis's statement: an invariant probabilistic measure for a group exists if and only if the quotient group does not contain a free subgroup with two generators (here is some specific subgroup of G defined in a canonical way). We also formulate and prove analogs of the Tits alternative for groups of homeomorphisms of the line.     

The Isomorphism Problem of Normalized Unit Groups of Group Algebras of a Class of Finite 2-groups

Let p be a prime and F_p be a finite field of p elements. Let F_pG denote the group algebra of the finite p-group G over the field F_p and V(F_pG) denote the group of normalized units in F_pG.Suppose that G and H are finite p-groups given by a central extension of the form■ and G’≌Z_p, m ≥ 1. Then V(F_pG)≌V(F_pH) if and only if G≌H. Balogh and Bovdi only solved the isomorphism problem when p is odd. In this paper, th...     

Unit Groups of Integral Finite Group Rings with No Noncyclic Abelian Finite p-Subgroups

It is shown that in the units of augmentation one of an integral group ring ? G of a finite group G, a noncyclic subgroup of order p ², for some odd prime p, exists only if such a subgroup exists in G. The corresponding statement for p = 2 holds by the Brauer–Suzuki theorem, as recently observed by Kimmerle.