Permuting operations on strings and their relation to prime numbers

Some length-preserving operations on strings only permute the symbol positions in strings; such an operation X gives rise to a family {Xn}n≥2 of similar permutations. We investigate the structure and the order of the cyclic group generated by Xn. We call an integer n X-prime if Xn consists of a single cycle of length n (n ≥ 2). Then we show some properties of these X-primes, particularly, how X-primes are related to X -primes as well as to ordinary prime numbers. HereX and X ′ range over well-known examples (reversal, cyclic shift, shuffle, twist) and some new ones based on Archimedes spiral and on the Josephus problem.