Symmetric Words in Dihedral Groups

Let G be a group and let w = w(x(1), x(2,) ..., x(n)) be a word in the absolutely free group F-n on free variables x(1), x(2), ..., x(n). The set S-(n) (G) of all words w such that the equality w(g(sigma 1), g(sigma 2), ..., g(sigma n)) = w(g(1), g(n), ..., g(n)) holds for all g(1), g(2), ..., g(n) is an element of G and all permutations sigma is an element of S-n is a subgroup of F-n, called the subgroup of n-symmetric words for G. In this paper, the groups S-(2) (D-p) and S-(3) (D-p) for dihedral groups D-p are determined, where p > 3 is a prime. In particular, it turns out that the groups S-(3) (D-p) are not abelian.