MHB Why Are There an Odd Number of Elements in a Finite Group Where g^3 Equals 1?

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Here is this week's POTW:

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Show that if $G$ is a finite group, then there are an odd number of elements $g\in G$ for which $g^3 = 1$.

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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

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This week's problem was solved correctly by castor28. You can read his solution below.

Spoiler

The elements of order $3$ can be grouped in pairs $\{g,g^{-1}\}$. Together with the identity, that makes an odd number of elements satisfying $g^3=1$.