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