Let $G$ be finite group, which has an even number of elements. Show that at least for two (distinct) elements $g,h$ of group $G$ one has $g*g = e$ and $h*h = e$.
I just started learning algebra and I have no ideas how I should solve this. I'm grateful for every explanation.
Reference: Fraleigh p. 48 Question 4.29 in A First Course in Abstract Algebra
Best Answer
Elementary way:
For $g$ simply take the identity $e$. To find another, assume that each element $h$ has an inverse $h^{-1}$ that is not $h$ ($h \neq h^{-1}$). Summing the elements $\{h, h^{-1} \}$ and $e$ up, you get an odd number of elements of the group. Contradiction. So there is another element $h$ such that $h = h^{-1}$, and you are done.
Alternative: just refer to Cauchy's Theorem.