Appears in collection : Combinatorics and Arithmetic for Physics : Special Days

We describe a group law on formal power series in non-commuting variables in- duced by their interpretation as linear forms on a Hopf algebra of sentences. We study the corresponding structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power se- ries that have been central in the context of non-commutative probability theory. Based on a joint work with K. Ebrahimi-Fard, N. Tapia and L. Zambotti.