Sequences of symmetry groups of infinite words

In this paper we introduce a new notion of sequence symmetry groups an infinite word. Given subgroup Gn the symmetric group Sn, it acts on set finite words length n by permutation. We associate to word w (Gn(w))n≥1 its groups: For each n, is Gn(w) Sn such that g(v) factor for permutation g∈Gn(w) and v w. study general properties characterize sequences several families words. show G there exists with Gn(w)=G. On other hand, structure possible quite restrictive: they cannot contain order certain cycles, transpositions some permutations. The can also generalized periodicity property. prove Sturmian more generally Arnoux-Rauzy are two large enough n; Toeplitz have exponential growth.