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I am looking at the automorphism group $G$ of a graph, represented as permutation matrices. The point in a proof I am trying to understand goes something like this:
"For any permutation matrix $P$ in $G$ there exists an orthogonal matrix $Q$ such that $Q^{-1}PQ=A$, where $A$ is a block-diagonal matrix representing a direct product of orthogonal groups. Hence there is an embedding of $G$ into this direct product."
Why is this? Is it always true that a morphism between two groups can be represented by a matrix $Q$ such that $Q^{-1}PQ=A$, where $P$ and $A$ are matrices representing elements of the domain and the image respectively? Or is this only the case for morphisms between two different representations of the same group?