I have a question about the standard notation for representing properties of permutations. This is best illustrated with a concrete example: let's take a permutation on 6 elements with cycle representation $\sigma=(134)(2)(56)$. We know that the cycle representation is unique. In the problem I'm working on, I need an abstract notation to refer to the set of cycles in the cycle representation of $\sigma$. In this example, this set is just $\{(134),(2),(56)\}$, but I don't know a technical name for this set and I imagine one exists. Could anyone point me to a reference which introduces standard notation of this type? So far I haven't been able to find one.
There are some other properties which I would like to give a name to. For example, once the cycle representation of a permutation is written down, we can talk about things like the set of elements acted on by a particular permutation. In my example above, the first cycle acts on the elements $\{1, 3, 4\}$, and it would be nice to have a standard notation for how to index into this subset of the original six elements.
Have you seen this (sort of thing)?
$(134)$ can be "described" by: $$\begin {pmatrix}1&2&3&4&5&6\\3&2&4&1&5&6\end {pmatrix}.$$
Because $1\to3,2\to2,3\to4,4\to1,5\to5,6\to6.$