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 Abstract: Let $\alpha$ be an automorphism of the totally disconnected group $G$. The compact open subgroup, $V$, of $G$ is \emph{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 $\siH$ 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 $\siH$ the analogues of eigenvalues. Acknowledgments: Research supported by A.R.C. Grant A69700321 Author information: School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia george@frey.newcastle.edu.au