On exponentials of exponential generating series

1: After identification of the algebra of exponential generating series with the shuffle algebra of ordinary formal power series, the exponential map exp! : XK[[X]] −→ 1 +XK[[X]] for the associated Lie group with multiplication given by the shuffle product is well-defined over an arbitrary field K by a result going back to Hurwitz. The main result of this paper states that exp! (and its reciprocal map log!) induces a group isomorphism between the subgroup of rational, respectively algebraic series of the additive group XK[[X]] and the subgroup of rational, respectively algebraic series in the group 1 +XK[[X]] endowed with the shuffle product, if the field K is a subfield of the algebraically closed field Fp of characteristic p.