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Let $G$ be an infinite group that is not simple. If $B$ is a subgroup of $G$, must $B$ necessarily be not simple as well?

I know that if $N$ is some proper normal subgroup of $G$, then $B\cap N$ is a normal subgroup of $B$. But we may also have that $B$ intersects trivially with $N$.

If no, is there a counter example? Would be grateful if anyone can enlighten me.

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Hint: Try $G = A_5 \times \mathbb Z$.

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