Classifying permutations in terms of their cycle notation

Question

Is there are a standard way of referring to permutations in terms of their cycle notation? For example: Does the set of all permutations in $S_4$ that can be expressed as the composition of two two-cycle permutations $\left\{ (12)(34), (13)(43), (14)(23)\right\}$ has a name?

Answer 1

The sets you're looking for are the conjugacy classes of $S_n$.

It's a good exercise to convince oneself that "$a$ and $b$ are conjugates in $S_n$" is equivalent to "$a$ and $b$ have the same cycle type".

Answer 2

Every permutation has a unique decomposition into disjoint cycles, and it isn't hard to show that this representation is unique. Once you have a permutation into disjoint cycles, you can talk about the cycle type (since it's unique). For example, we might say $(14)(235)(67) \in S_7$ has cycle type 322 (order doesn't matter).

The cycle type is preserved under conjugation. That is, the permutations $w$ and $\sigma w\sigma^{-1}$ have the same cycle type, for any $\sigma$.