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If $G$ is a group of odd order, prove for any nonidentity element $x \in G$ that $x$ and $x^{-1}$ are not conjugate in $G$....

Question

If $G$ is a group of odd order, prove for any nonidentity element $x \in G$ that $x$ and $x^{-1}$ are not conjugate in $G$.

If $G$ is a group of odd order, prove for any nonidentity element $x \in G$ that $x$ and $x^{-1}$ are not conjugate in $G$.



Answers

If $ f $ and $ g $ are both even functions, is the product $ fg $ even? If $ f $ and $ g $ are both odd functions, is $ fg $ odd? What if $ f $ is even and $ g $ is odd? Justify your answers.


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