  Let \alpha be an automorphism of the totally disconnected group G. The
compact open subgroup, V, if G is tidy for \alpha if [\alpha(V') :
\alpha(V')\cap V'] is minimised at V, where V' ranges over all compact open
subgroups of G. Identifying a subgroup tidy for \alpha is analogous to
identifying a basis which puts a linear transformation into Jordan canonical
form. This analogy is developed here by showing that commuting automorphisms
have a common tidy subgroup of G and, conversely, that a group H of
automorphisms having a common tidy subgroup V is abelian modulo the
automorphisms which leave V invariant. Certain subgroups of G are the analogues
of eigenspaces and corresponding real characters of H the analogues of
eigenvalues.
