  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 $\lambda_{1}$ and $\lambda_{2}$ 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 $\lambda_{1}$ and $\lambda_{2}$ 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 $\lambda_{1}$ and $\lambda_{2}$ interact with finite group
presentations. This is of interest independent of the study of the quantities
$\lambda_{1},\; \lambda_{2}$. Finally, we discuss several conjectures and open
questions for future consideration.
