Generalizations of Commutativity in Dihedral Groups

Authors

Noah A. Heckenlively, Rose Hulman Institute of TechnologyFollow

Abstract

The probability that two elements commute in a non-Abelian finite group is at most 5 8 . We prove several generalizations of this result for dihedral groups. In particular, we give specific values for the probability that a product of an arbitrary number of dihedral group elements is equal to its reverse, and also for the probability that a product of three elements is equal to a permutation of itself or to a cyclic permutation of itself. We also show that for any r and n, there exists a dihedral group such that the probability that a product of n elements is equal to its reverse is r q for some q coprime to r, extending a known result.

Author Bio

Noah Heckenlively developed this mathematics during his Senior year as part of his Senior Thesis at Rose Hulman Institute of Technology. He has now gone on to graduate school in the mathematics program at Auburn. After getting a Master's, he anticipates going into industry as a Data Scientist.

Faculty Sponsor

Tom Langley

Recommended Citation

Heckenlively, Noah A. (2022) "Generalizations of Commutativity in Dihedral Groups," Rose-Hulman Undergraduate Mathematics Journal: Vol. 23: Iss. 2, Article 3.
Available at: https://scholar.rose-hulman.edu/rhumj/vol23/iss2/3