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Im having trouble with the second part of this question,

Let $H$ be a normal subgroup of $G$ with $|G:H| = n$,

i) Prove $g^n \in H$ $\forall g \in G$ (which i have done)

ii) Give an example to show that this all false when $H$ is not normal in $G$.(which I am having trouble with showing)

Any suggestions?

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2 Answers 2

Hint: If $H$ is not normal in $G$, then $G$ is necessarily nonabelian. Consider the smallest nonabelian group you know.

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See my answers here If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?. You can take $H$ to be any non-normal Sylow $p$-subgroup of $G$.

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