  We study a particular group law on formal power series in non-commuting
variables induced by their interpretation as linear forms on a suitable graded
connected word Hopf algebra. This group law is left-linear and is therefore
associated to a pre-Lie structure on formal power series. We study these
structures and show how they can be used to recast in a group theoretic form
various identities and transformations on formal power series that have been
central in the context of non-commutative probability theory, in particular in
Voiculescu's theory of free probability.
