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I need a counterexample to the following statement: if A, B are subgroups of a group G, and A is a normal subgroup of B, B is a normal subgroup of G, then A is a normal subgroup of G. I've tried using groups such as $Z/nZ$ and $Z_n^{*}$, but neither seem to work. Thanks for your help.

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marked as duplicate by Stahl, Pragabhava, Watson, drhab, Prahlad Vaidyanathan Oct 21 '16 at 18:42

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Consider $A_4$. We have $\Bbb{Z}_2 \unlhd V_4 \unlhd A_4$ but $\Bbb{Z}_2$ is non-normal in $A_4$

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