Aspects of Infinite Groups

The following sections are included:

• PREFACE

• CONTENTS

An algebraic structure arising in the formulation of a lightweight key agreement protocol yieldsa new concept of commutator whose identities are quite traditional

No abstract received.

Recently there has been an active line of research on noncommutative algebraic cryptography. This involves the use of noncommutative algebraic objects as the platforms for encryption systems. Most of this work, such as the Anshel-Anshel-Goldfeld scheme, the Ko-Lee scheme and the Baumslag-Fine-Xu Modular group scheme use nonabelian groups as the basic algebraic object. Some of these encryption methods have been successful and some have been broken. It has been suggested that at this point further pure group theoretic research, with an eye towards cryptographic applications, is necessary. In the present study, the start of large project, we discuss various methods for using noncommutative algebraic objects as methods to develop cryptographic schemes and cryptoprimitives. The project has several parts; the development of general algebraic schemes for cryptoprimitives, the implementation of appropriate platforms for these schemes and finally an analysis of the potential security of the schemes.

We provide a detailed structure description of the derived subgroups of the free nilpotent groups of finite rank. This description is then applied to computing the nonabelian tensor squares of the free nilpotent groups of finite rank.

In this note we consider a free product G of finitely many cyclic groups of finite or infinite order and develop an explicit and straightforward recurrence formula for the number of free subgroups of G which includes only the given group theoretic data as the number of the free product factors and the orders of the given cyclic groups.

The object of this note is to give a concise proof of the following theorem: Let G = B(m,n) = 〈a, b; a^-1b^ma = bⁿ|m ≠ 0, n ≠ 0, m, n ∈ ℤ〉 and H = B(m′,n′) = 〈x,y; x^-1y^m′ x = y^n′|m′ ≠ 0, n′ ≠ 0, m′, n′ ∈ ℤ〉. Then G ≅ H if and only if m = m′ and n = n′.

In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) we study finitely generated limit algebras of .

No abstract received.

This paper proposes the Tarski Problem for free groups in a Burnside variety, B_n, where n is a sufficiently large odd integer so that Adian's results hold. We note that just as in the case of absolutely free groups it is easy to show that the nonabelian free groups in B_n for n as above are all universlly equivalent.

Let F_n be a free group of rank n with basis X = {x₁, x₂, ⋯, x_n}. If Y = {y₁, y₂, ⋯, y_n} is any other set of generators for F_n then they form a basis for F_n since finitely generated free groups are Hopfian. In this paper which shall show that this fact characterizes finitely generated free groups.

We show that if A is a k × n matrix over a principal ideal ring R, with k < n, and if d is any element of the ideal generated by the k × k minors of A, then A forms the top k rows of an n × n matrix of determinant d. This parallels a 1981 result of Gustafson, Moore, and Reiner, and continues a program initiated by Hermite in 1849. Then we use these results to obtain an extension of a 1997 result of Richter and Wardlaw for good matrices.

These are expanded lecture notes of a series of expository talks surveying basic aspects of group cohomology and homology. They were written for someone who has had a first course in graduate algebra but no background in cohomology. You should know the definition of a (left) module over a (non-commutative) ring, what ℤ[G] is (where G is a group written multiplicatively and ℤ denotes the integers), and some ring theory and group theory. However, an attempt has been made to (a) keep the presentation as simple as possible, (b) either provide an explicit reference or proof of everything.

Several computer algebra packages are used to illustrate the computations, though for various reasons we have focused on the free, open source packages, such as GAP [Gap] and SAGE [St] (which includes GAP). In particular, Graham Ellis generously allowed extensive use of his HAP [E1] documentation (which is sometimes copied almost verbatim) in the presentation below. Some interesting work not included in this (incomplete) survey is (for example) that of Marcus Bishop [Bi], Jon Carlson [C] (in MAGMA), David Green [Gr] (in C), Pierre Guillot [Gu] (in GAP, C++, and SAGE), and Marc Röder [Ro].

Though Graham Ellis' HAP package (and Marc Röder's add-on HAPcryst [Ro]) can compute comhomology and homology of some infinite groups, the computational examples given below are for finite groups only.

In this paper we study residual solvability of doubles of residually solvable groups. We find suitable conditions where this kind of structure is residually solvable, and show that in general this is not the case. However this kind of structure is always meta-residually-solvable.

No abstract received.

No abstract received.

Given a nilpotent group G and a prime p, there is a unique p-local group G_(p) which is, in some sense, the "best approximation" to G among all p-local nilpotent groups. G_(p) is called the p-localization of G. Let I A(G) be the group of automorphisms of G that induce the identity on G/[G, G]. I A(G) turns out to be nilpotent so its p-localization exists. Two groups are said to be in the same localization genus if their p-localizations are isomorphic for all p. We prove that if two finitely generated, torsion-free nilpotent, and metabelian groups lie in the same localization genus, their I A-groups also lie in the same localization genus. The method of proof involves basic sequences and commutator calculus.