It's a home work problem I got:
Find $4$ different subgroups of $S_4$ isomorphic to $S_3$ and $9$ isomorphic to $S_2$.
My approach is: since $S_3=\{1, (123),(132),(12),(23),(13)\}$, just take the groups of permutations on $\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\}$, obviously they are all subgroups of $S_4$.
To find the isomorphism, for each subgroup, just assign 1 to the first element, 2 to the second, 3 to the third. This is going to be an isomorphism.
For example, in the group of permutations on $\{1,2,4\}$, assign $(124)\rightarrow(123) (24)\rightarrow(23),(142)\rightarrow(132), (14)\rightarrow(13)$, etc.
I think this method is fine, but I have trouble with the second part of the problem. That is, to find the $9$ different subgroups of $S_4$ isomorphic to $S_2$. When I pick $2$ elements out of $\{1,2,3,4\}$, there can only be $\frac{4!}{2!2!}=6$ ways, which means this method only gives $6$ different subgroups isomorphic to $S_2$, but the problem says there are $9$.
Is my method wrong? Or are there some other subgroups that I've missed?
Thanks!!!
Best Answer
Here is a recipe for finding all subgroups of Sn that are isomorphic to a fixed and completely understood group G, like G = S3.
First, find all conjugacy classes of subgroups H of G, and label them by their index $[G:H] = |G| / |H|$. For example, for G = S3:
Each of these defines a way for G to act (multiplication on the cosets of H in G). For instance, for G = S3:
Now write n as an unordered sum of the indices. For instance, if n = 4, then there is only one way:
A better example is n = 5, where there are two ways:
Maybe even better is n = 6, where there are four ways:
In each case, we are writing a general action of G as a sum of simple "transitive" actions of G on the cosets of a subgroup H. This is called the orbit stabilizer theorem; your course should mention this at least superficially. The idea of literally adding them up is one way to view "permutation characters" in character theory; that will be in a later course, most likely.
This only works if (a) you understand G very well, and (b) the "big" group is Sn. A similar thing works if the "big" group is a general linear group, but then the H are replaced by modules and this is called representation theory of finite groups.