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Are normal subgroups transitive?

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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/nZZ/nZ and Z∗nZ_n^{*}, but neither seem to work. Thanks for your help.

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1 Answer

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Consider A4A_4. We have Z2⊴V4⊴A4\Bbb{Z}_2 \unlhd V_4 \unlhd A_4 but Z2\Bbb{Z}_2 is non-normal in A_4A_4