A computational study of the factor groups of the lower central series of a certain free product.
详细信息
文摘
Let G denote the free product of three cyclic groups of order 2. The purpose of this study is the computational investigation of the factor groups of the lower central series of G. The motivation for studying this particular example is twofold. First, we wish to gain better insight into the performance of algorithms that have emerged over the past two decades in computational group theory. Further we would like to contribute to better understanding of free products of cyclic groups and their lower central series by learning more about these specific examples.;We begin by introducing certain fundamental concepts from the commutator calculus and demonstrate how they can be used in computing presentations of the factor groups of the lower central series of G.;Next, we present the results of computational experiments with the finite nilpotent factors of the lower central series of G. We introduce the concept of supergirth. We examine the Cayley graphs of these factor groups and generate conjectures based on our computations about both their supergirths and diameters. We identify the factor group $G/G\sb3$ as a known nilpotent group of order 64.;We continue the examination of the group G by using the classical representation of G as a subgroup of the modular group (the group PSL(2, Z) of linear fractional transformations). This allows us to relate our study of G with the concepts of the trace function, the level function and the parabolic class number, all of which are integral to the study of the modular group $\Gamma.$ We prove a formula for the level of the terms of the lower central series of G in $\Gamma.$;In the final chapter we formulate a hybrid algorithm for computing free bases for the terms of the lower central series of G using two well-known rewriting processes and specify a power commutator presentation for each of the factor groups $G/G\sb{n}.$