Example of A, B, G such that A is a normal subgroup of B, B is a normal subgroup of G, but A is not a normal subgroup of G [duplicate]

Question

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• Are normal subgroups transitive? 3 answers

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.

Answer 1

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