The main result to be discussed is a non-commutative
version of the following linear algebra result: if $T$ is a linear
transformation of order $n$ ($T^n=Id$) on a vector space $V$,
then $T$ diagonalizes and the eigenvalues are the $n$-th roots of unity.
In the non-commutative version, the role of V is played by the group
of characters on a graded connected Hopf algebra, and the role of $T$ by
a canonical automorphism associated to the grading. These notions will
be defined and illustrated with several examples of a combinatorial nature.
The factorization corresponding to the Hopf algebra of quasi-symmetric
functions will be one of the main focuses. It involves interesting
combinatorial constructions and allows for various applications.