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Assume $\langle G,*\rangle$ is a finite group, try to prove that the amount of all elements with order bigger than $2$ is even.

I understand that the unit element is unique and order-$2$ element has: $a^{-1}=a$

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You can think of the inverse operation as a Z/2 action on the set and its only fixed points are the elements with order <= 2. Everything else is in an orbit with two elements and what you want follows – user148177 Jun 3 '14 at 11:22
up vote 2 down vote accepted

Continuing on your good start, check that for elements of order greater than 2 the element and its inverse are different (also keep in mind that the inverse of the inverse of an element is the element itself), and continue from there.

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