$H = \left\{ id, (12)(34), (13)(24), (14)(23)\right\}$ is a subgroup of $A_4$. A normal subgroup is a subgroup where the right and left cosets are the same. To show that $H$ is a normal subgroup would I have to compute every single coset and show that the left and right cosets are the same? Is there a clever way to do this problem?
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I don't think $H$ is a subgroup of $A_4$, since $|H| = 7$, $|A_4| = 12$, but $7$ does not divide $12$, contradicting Lagrange's theorem. Maybe you put some extra element there. If were $|H| = 6$ instead of $7$, then $H$ would have index $12/6 = 2$, and every subgroup with index $2$ is automatically normal. – Ivo Terek Nov 06 '16 at 18:01
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Woops, I incorrectly typed the subgroup. I have fixed it in the edit. – MomoTheSir Nov 06 '16 at 18:12