# example of an inﬁnite group [closed]

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 $$n$$

## closed as off-topic by Derek Holt, TheSimpliFire, Xander Henderson, Shaun, Moishe KohanSep 21 at 17:51

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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?)