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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.

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$A_5$ is simple so you will not find any interesting normal subgroups. Let me give you a hint: is $g,a,b \in G$, then $g^{-1}abg=g^{-1}ag \cdot g^{-1}bg$. This should help you proving that the subgroup generated by a conjugacy class is actually normal.

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  • $\begingroup$ Thanks, I managed to prove it now. I specifically chose a simple group because I thought that the statement was false, so if the generated subgroup were to be normal it could only be trivial. My hypothesis was that the generated subgroup could have been proper. $\endgroup$
    – user305860
    Dec 8 '16 at 6:59
  • $\begingroup$ Erik, yes well done!!! $\endgroup$ Dec 8 '16 at 9:04
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A subgroup generated by conjugacy classes is always normal.

Hint: Let $N$ be a subgroup generated by $<g^x|x\in G>$. Take a $n\in N$.

Then $n=h_1h_2...h_k$ for $h_i=g^x$ for some $x\in G$. Now what can you say about $n^y$ for some $y\in G$ ?

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