Is a subgroup generated by a conjugacy class normal?
I have been trying to look for a counter-example. Naturally, I thought that simple groups would be suitable. However, the generated subgroup seems to always be the whole group itself, hence trivially normal. I have tried this with $A_5$.
Also, this tells us that the generated subgroup would be the normal closure, which intuitively sounds too strong. I have tried proof by contradiction, both from the positive and negative position, but it leads me nowhere.
Any hints and directions are appreciated.