The Power of Group Generators and Relations: An Examination of the Concept and Its Applications ()
ABSTRACT
This paper investigates the approach of presenting
groups by generators and relations from an original angle. It starts by
interpreting this familiar concept with the novel notion of “formal words” created by juxtaposing
letters in a set. Taking that as basis, several fundamental results related to
free groups, such as Dyck’s Theorem, are proven. Then, the paper
highlights three creative applications of the concept in classifying finite
groups of a fixed order, representing all dihedral groups geometrically, and
analyzing knots topologically. All three applications are of considerable
significance in their respective topic areas and serve to illustrate the
advantages and certain limitations of the approach flexibly and
comprehensively.
Share and Cite:
Zhou, T. (2018) The Power of Group Generators and Relations: An Examination of the Concept and Its Applications.
Journal of Applied Mathematics and Physics,
6, 2425-2444. doi:
10.4236/jamp.2018.611204.