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I am trying to look at the alternating group $A_4$ and struggling to come up with any nice proofs and was wondering if anyone could help. I am trying to prove that $A_4$ as a subgroup of $S_4$ is a normal subgroup and that the Klein four subgroup is a normal subgroup of $A_4$. I know these can be done by multiplying out and proving left and right corsets coincide but am sure there must be a more elegant solution. Any help would be much appreciated thanks.

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Prove that $A_4$ is a normal subgroup of $S_4$.

There is nothing special about $4$ here; it is true more generally that $A_n$ is a normal subgroup of $S_n$ for any $n$. As with many questions, yours is easy enough to answer if you reformulate it correctly. We would like to show that if $\sigma$ is a permutation in $A_4$ and $\tau$ is any permutation of $S_4$ then $\tau\sigma\tau^{-1}$ is a permutation of $A_4$. Two questions:

  1. What is the definition of $A_4$?
  2. Does the permutation $\tau\sigma\tau^{-1}$ satisfy the definition characterizing elements of $A_4$?

Show that $A_4$ contains a normal subgroup isomorphic to the Klein four group.

Your second question is much more computational than the previous one. I recommend looking for elements of order $2$ in $A_4$, as every element of the Klein group besides the identity has order $2$.

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For the first part observe that $[S_4:A_4]=\frac{24}{12}=2$ Thus $A_4$ is normal in $S_4$. For the second part see this.

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