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I try to show that $[A_4,A_4] \subset V_4$.

Just by trial and error I found that $(12)(34)=[(123),(124)]$. I think I can find the other two also by trial and error, but is there also a smarter method to find those commutators ?

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  • $\begingroup$ The suggested edit by @CristianBaeza is not right. Do you know of the superset symbol? $\endgroup$ – 6005 Aug 23 '16 at 23:25
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$V_4$ is normal in $A_4$ since it is a union of two conjugacy classes. Now note that $A_4/V_4 \cong \mathbb{Z}/3$ is abelian. Therefore, $V_4$ contains the commutator subgroup.

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  • $\begingroup$ sorry ! I meant $[A_4,A_4] \subset V_4 $ ! $\endgroup$ – Kasper May 4 '13 at 22:02

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