Exercise 29 of Section 4 showed that every finite group of even order 2n contains an element of order 2. Using the theorem of Lagrange, show that if n is odd, then an abelian group of order2n contains precisely one element of order 2.

Data from exercise 29 of section 4

Show that if G is a finite group with identity e and with an even number of elements, then there is a ≠ e in G such that a* a = e