# subgroup structure of $S_4$

In the list of Young subgroups of $S_4,$ we find $\langle(12)\rangle, \langle(13)\rangle, \langle(14)\rangle, \langle(23)\rangle, \langle(24)\rangle, \langle(34)\rangle,$ but we don't find $\langle(12)(34)\rangle, \langle(13)(24)\rangle,\langle(14)(23)\rangle,$ while they are all isomorphic to $S_2.$ I'm confused.

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What is a young subgroup? – Chris Eagle May 13 '12 at 10:10
@ChrisEagle: encyclopediaofmath.org/index.php/Young_subgroup – Isaac Solomon May 13 '12 at 10:23

A Young subgroup is the direct product of the symmetric groups on the components of the partition. While all these groups are abstractly isomorphic to $S_2$, only the first batch you list is actually $S_2$ on a two-element subset of $\{1,2,3,4\}$, whereas e.g. in the first example of the second batch the group would also have to include $(12)$ and $(34)$ separately in order to be the Young subgroup for the partition $\{1,2,3,4\}=\{1,2\}\cup\{3,4\}$. You can't write down a partition such that $\langle(12)(34)\rangle$ contains all combinations of all permutations on all subsets forming the partition.