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I am studying group theory,

I want to an example of an infinite group, say, $G$, such that $G$ contains a normal subgroup $H$ and $Ord(aH) = n$ in $G/H$ but $G$ does not contain an element of order $n$

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Let $G = Z$ under normal addition, and $n = 3$, and $H = 3Z$.

Then $H$ is normal in $Z$ (Easy to prove)

and $Ord(1+3Z) = 3$ (find yourself), but $Z$ does not contain an element of order $3$ (I hope you know why?)

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