On Primitive Words
Salwa Bouall, Mongi Naimi · 2010
Let A be a flnite alphabet, A ⁄ be the set of all flnite words. Let c : A ⁄ ! A ⁄ be the circular shift deflned by c A () = and c A (at) = ta, for each a 2 A and t 2 A ⁄ ). Then the additive group (Z;+) acts on A ⁄ by the action Z £ A ⁄ i! A ⁄ , which sends the pair (k;u) to the word c k (u). In this paper, we prove that the stabilizer of a nonempty word u of length n is exactly Su = ‚Z := fk‚ j k 2Zg, where ‚ is the length of the primitive root of u. Using Burnside’s counting orbit theorem, we give an alternative proof of the total number of necklaces of length n on k symbols: N(n;k) = 1