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Recently I have seen a post whose link is the following.

I am not able to prove the first statement, namely, "A group $G$ is simple if and only if for any $1 \neq x \in G$, the conjugacy class of $x$ in $G$ generates the whole group $G$."

Can anyone tell me or give me some references where I can find its proof?

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If the conjugacy classes of some $x$ generate a strict subgroup $H\subset G$, can you show that $H$ is a normal subgroup?

And conversely, if $H\subset G$ is a normal subgroup, can you show that for any $x\in H$, its conjugates generate a strict subgroup of $G$?

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