So I know that the Klein group is the group with 4 elements that is not cyclic but I'm stuck from there onwards?
[Math] Find a subgroup of $S_4$ that is isomorphic to V, the Klein group.
finite-groupsgroup-theory
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Best Answer
Here's one way of going about this:
Every non-identity element in $V_4$ is of order $2$. Therefore, a good starting point would be to choose your favorite transposition in $S_4$. Call it $\pi$. Next, you could choose another transposition and call it $\sigma$. However, we want to ensure that the composition of $\pi$ and $\sigma$ also has order $2$. This will only happen if $\pi$ and $\sigma$ are disjoint transpositions (why?).
Finish up by considering the subgroup generated by $\pi$ and $\sigma$.