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?

I think just the cycles themselves have the standard name "cycle". — Ethan Bolker, Oct 26 '16 at 19:54
Every permutation can be decomposed into a product of disjoint cycles.
http://math.stackexchange.com/questions/319979/how-to-write-permutations-as-product-of-disjoint-cycles-and-transpositions

Every cycle, in turn, is a product of transpositions. — user8960, Oct 26 '16 at 19:54
@EthanBolker: So, can I call (123)(456) a (3,3)-cycle permutation and (12)(3456) a (2,4)-cycle permutation? — Sergio Parreiras, Oct 26 '16 at 19:58
Yes, you could, but it's not standard notation. @HenningMakholm 's accepted answer is the way to go. — Ethan Bolker, Oct 26 '16 at 23:55

score 2 · Accepted Answer · answered Oct 26 '16 at 20:05

2

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".

answered Oct 26 '16 at 20:05

hmakholm left over Monica

286,031

score 1 · Answer 2 · edited Oct 26 '16 at 20:06

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 $\sigma$.

Classifying permutations in terms of their cycle notation

2 Answers2