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Let $G$ be a group that has an element $g$ with exactly two conjugate elements.Prove that $G$ contains a proper non trivial normal subgroup $N$.

Can someone give some help with this?

Thank you in advance!

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    Exactly two, including itself? – 2017-02-26
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    i think that the problem asks about two different elements from g – 2017-02-26
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    If it says exactly two conjugate elements then that is exactly what it means. Why would you imagine that this would not include $g$ itself? Every element is conjugate to itself. But as it happens, the conclusion would still hold if $g$ had exactly three conjugates, or even exactly four conjugates, or five even. The smallest conjugacy class of $A_5$ has size $12$. – 2017-02-26
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    @Derek I see how to prove that (with a permutation representation) but this problem is a lot easier. – 2017-02-26

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The size of a conjugacy class is the index of the centralizer of the element, and a subgroup of index $2$ is normal.

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    this is true but i also have to prove that the centralizer is non trivial and a proper subgroup of $G$.. – 2017-02-26
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    @capo a subgroup of index $2$ has half as many elements as the group. Thus it is proper, and since this can't happen in a group of order $2$ it is also nontrivial. – 2017-02-26
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    O yes..thank you for your help. – 2017-02-26
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    @capo No problem. – 2017-02-26