Infinite Group Theory

The following sections are included:

• Introduction
• Contents

Let G be a group with the weak minimal condition on non-permutable subgroups. The main results are as follows. If G is locally finite then either G is quasihamiltonian or G is Chernikov; if G is a generalized radical group then G is either quasihamiltonian or soluble-by-finite of finite rank.

This paper serves as an introduction to secret sharing scheme, and it provides the fundamental understandings to the scheme from various aspects. We first review the basics of a Shamir threshold scheme, and discuss various enhancements so that the scheme can be proactive and verifiable. We then show how a Shamir scheme can be extended to realize any general access structure. We also point out the relationship between a Shamir scheme and other topics such as error correction code, ramp scheme, information disposal algorithm and multiparty computation. Finally, we briefly discuss other platforms for its implementation.

It is well-known that the direct product of left-orderable groups is left-orderable and that, under a certain condition, the semi-direct product of left-orderable groups is left-orderable. We extend this result and show that, under a similar condition, the Zappa-Szep product of left-orderable groups is left-orderable. Moreover, we find conditions that ensure the existence of a partial left and right invariant ordering (bi-order) in the Zappa-Szep product of bi-orderable groups and prove some properties satisfied.

For each Baumslag-Solitar group BS(m, n) (m, n ∊ 𝕫 \ {0}), a totally disconnected, locally compact group, G_{m, n}, is constructed so that BS(m, n) is identified with a dense subgroup of G_{m, n}. The scale function on G_{m, n}, a structural invariant for the topological group, is seen to distinguish the parameters m and n to the extent that the set of scale values is

Given a nonabelian commutative transitive (CT) group G, we show that the elementary theory of G is axiomatizable by the set H(G) of Horn sentences true in G togeher with CT. We also obtain results about the elementary theory of a group N free in the variety of all groups nilpotent of class at most c. Finally for a nonabelian CSA group G we show that its universal theory is axiomatizable by the set Q(G) of quasi-identities true in G together with CT and the diagram of G. This last result generalizes a result of Myasnikov and Remeslennikov.

A group is G commutative transitive or CT if commuting is transitive on nontrivial elements. A group G is CSA or conjugately separated abelian if maximal abelian subgroups are malnormal. These concepts have played a prominent role in the studies of fully residually free groups, limit groups and discriminating groups. They also play a role in the solution to the Tarski problems. CSA always implies CT however the class of CSA groups is a proper subclass of the class of CT groups. For limit groups and finitely generated elementary free groups they are equivalent. In this paper we survey the relationship between the two concepts and also examine a series of related results. In [FGRS 2] it was proved that a finite CSA group must be abelian and if G is CT then G is not CSA if and only if G contains a nonabelian subgroup G₀ which contains a nontrivial abelian subgroup H that is normal in G₀. In [FGRS] it was shown that the groups PSL(2, K) for K a field are crucial in studying the relationship between CT and CSA. PSK(2, K) is never CSA and here we show when it is CT.

We characterize the universal theory of the free Burnside groups of large prime exponent and briefly show how to extend and modify our results to arbitrary sufficiently large odd exponent.

The problem of determining whether or not a non-elementary subgroup of PSL(2,ℂ) is discrete is a long standing one. The importance of two generator subgroups comes from Jørgensen’s inequality which has as a corollary the fact that a non-elementary subgroup of PSL(2,ℂ) is discrete if and only if every non-elementary two generator subgroup is. A solution even in the two-generator PSL(2,ℝ) case appears to require an algorithm that relies on the concept of trace minimizing that was initiated by Rosenberger and Purzitsky in the 1970’s. Their work has lead to many discreteness results and algorithms. Here we show how their concept of trace minimizing leads to a theorem that gives bounds on the hyperbolic lengths of curves on the quotient surface that are the images of primitive generators in the case where the group is discrete and the quotient is a pair of pants. The result follows as a consequence of the Non-Euclidean Euclidean algorithm.

Probabilistic reasoning under the principle of maximum entropy (so-called MaxEnt principle) is a viable and convenient alternative to graph-based methodologies such as Bayesian networks that realises an idea of information economy, i.e., of being as unbiased as possible. For relational conditional knowledge, the aggregating semantics provides a semantic framework which sorts well with MaxEnt. In this paper, we exploit the group structure that underlies relational conditional knowledge bases to condense both, the aggregating semantics and the MaxEnt formalism. Based on this group structure, we present an approach to performing MaxEnt reasoning that makes use of symbolic computations. Given a knowledge base, we encode the MaxEnt optimization problem into a system of polynomial equations and then apply elimination theory to find MaxEnt inferences as points of an affine variety.

The authors consider some new approaches to study of infinite-dimensional linear groups.

The paper discuss the relationship between the factors of the upper and lower series in the Lie rings. As a result some analogs of well known important group theoretical results were enhanced to Lie rings.

We show a transfer result from individual groups to wreath products. Namely, we prove that the conjugacy problem in the wreath product A ≀ B of two groups A and B is log-space decidable, provided the factor groups A and B both have log-space decidable conjugacy problem and B has log-space computable power problem. If, additionally, A and B have bounded torsion and A has log-space computable power problem, we show that the iterated wreath product A ≀ⁿ B also has log-space decidable conjugacy problem. We apply these general results to show that free solvable groups and wreath products of two abelian groups (in particular, the lamplighter group, ℤ₂ ≀ ℤ) have log-space decidable conjugacy problem.

The purpose of this paper is to examine the relationship between groups, group presentations, their Cayley graphs, and associated Markov Processes. In particular, we prove that for a finite group derived from an ergodic Markov Process, a process we describe in the paper, the long range equilibrium vector is uniform on the group elements, as to be expected. We also prove a theorem giving a complete characterization of finitely generated free groups in terms of their associated Markov Processes.