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Problem 95 (Subgroups And Normality)

Suppose that $G$ is a group and that $A\leq B\leq G$.

  1. If $A$ is normal in $G$, prove that $A$ is normal in $B$.
  2. If $A$ is normal in $B$ and $B$ is normal in $G$, must $A$ be normal in $G$? Find a counter example to this.

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