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 {X_n}_n≥2 of similar permutations. We investigate the structure and the order of the cyclic group generated by X_n. We call an integer n X-prime if X_n 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. Here X and X^′ range over well-known examples (reversal, cyclic shift, shuffle, twist) and some new ones based on the Archimedes spiral and on the Josephus problem.