On the group of a rational maximal bifix code

We give necessary and sufficient conditions for the group of a rational maximal bifix code Z to be isomorphic with the F-group of Z boolean AND F, when F is recurrent and Z boolean AND F is rational. The case where F is uniformly recurrent, which is known to imply the finiteness of Z boolean AND F, receives special attention. The proofs are done by exploring the connections with the structure of the free profinite monoid over the alphabet of F.