# Characterization of simple groups in terms of its conjugacy classes [closed]

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?

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 $$G$$?