The special subgroup of invertible non-commutative rational power series   as a metric group
  We give an easy proof of Sch\"utzenberger's Theorem stating that non-commutative formal power series are rational if and only if they are recognisable. A byproduct of this proof is a natural metric on a subgroup of invertible rational non-commutative power series. We describe a few features of this metric group. 
Roland Bacher (IF)