Document Type
Article
Publication Date
4-20-2017
Publication Title
Discrete Applied Mathematics
Abstract
Given a finite group with a generating subset there is a well-established notion of length for a group element given in terms of its minimal length expression as a product of elements from the generating set. Recently, certain quantities called λ1 and λ2 have been defined that allow for a precise measure of how stable a group is under certain types of small perturbations in the generating expressions for the elements of the group. These quantities provide a means to measure differences among all possible paths in a Cayley graph for a group, establish a group theoretic analog for the notion of stability in nonlinear dynamical systems, and play an important role in the application of groups to computational genomics. In this paper, we further expose the fundamental properties of λ1 and λ2 by establishing their bounds when the underlying group is a dihedral group. An essential step in our approach is to completely characterize so-called symmetric presentations of the dihedral groups, providing insight into the manner in which λ1 and λ2 interact with finite group presentations. This is of interest independent of the study of the quantities λ1,λ2. Finally, we discuss several conjectures and open questions for future consideration.
Keywords
Dihedral groups, Generating set, Group presentations, Minimal length, Word length
Volume
221
First Page
33
Last Page
45
DOI
10.1016/j.dam.2017.01.002
ISSN
0166218X
Recommended Citation
Allocca, Michael P.; Graham, Jason M.; Price, Candice R.; Talbott, Shannon N.; and Vasquez, Jennifer F., "Word Length Perturbations in Certain Symmetric Presentations of Dihedral Groups" (2017). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/120
Comments
Peer reviewed accepted manuscript.