I asked a question earlier today if someone can help me understanding conjugacy classes. There we discussed a view aspects and now I wanted to list the conjugacy classes of $S_4$ with their cardinality, but I somehow get stuck. I have the following at the moment:
We know that $S_4=Bij(\{1,2,3,4\},\{1,2,3,4\})$ and that $|S_4|=4!=24$. Now let us first compute all partitions of 4 and we remark that we have 5 of them, indeed $$4=1+1+1+1,\,\,\,\,\,\,\,\, 4=2+1+1,\,\,\,\,\,\,\,\, 4=3+1,\,\,\,\,\,\,\,\, 4=4, \,\,\,\,\,\,\,\,4=2+2$$ Since we know that the set of conjugacy classes in $S_4$ is isomorphic to the cardinality of the set of partitions of $4$ we know that we need to consider 5 conjugacy classes.
- $4=1+1+1+1$, this correxponds to the conjugacy class of $id\in S_4$ but since the identity commutes with all elements we get that $Cl(id)=\{id\}$ and thus $|Cl(id)|=1$
- $4=2+1+1$, this corresponds tho the conjugacy class of a transposition
I don‘t see how to compute point 2), first why exaclty is it the conjugacy class of a transposition, so I mean with the partition one see that our $\sigma$ has a decomposition of a transposition and two "fixed points" but is it because of that that one only needs to consider the conjugacy class of the transposition and can ignore the two "fixed points"? Furthermore how do I get the cardinality?
Thanks for your help.