# Normal Subgroup Counterexample [closed]

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?

## closed as off-topic by user26857, daw, user91500, apnorton, user98602 May 12 '15 at 15:08

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• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – user26857, daw, user91500, apnorton, Community
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Hint: If $H$ is not normal in $G$, then $G$ is necessarily nonabelian. Consider the smallest nonabelian group you know.
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 $G$.