Suppose $\sigma$ and $\tau$ are permutations such that $\sigma(x)\not=x\implies \sigma(x)=\tau(x)$. Intuitively, I would like to think of $\sigma$ as a restriction (or projection) of $\tau$ onto a restricted subspace. Another way of phrasing this: the cycle of structure of $\sigma$ is "contained" in the cycle structure of $\tau$.
However, I do not know the proper terminology to express this. I would like to write something like $\sigma \subset \tau$, without being misleading. Can anyone suggest the proper terminology or notation for this situation?
