Suppose that G is an Abelian group with an odd number of elements. show that the product of all the elements of G is the identity.
Let |G|=8. Show that G must have and element of order 2.
Hint: Pair $\displaystyle x$ and $\displaystyle y$ so that $\displaystyle xy=e$. Note why you need the fact that $\displaystyle |G|$ is odd.
Let $\displaystyle a\in G - \{e \}$ and construct $\displaystyle \left< a\right>$. By Lagrange's theorem $\displaystyle |\left< a \right>| = 2,4,8$. Since this is a cyclic group and 2 divides its order we can find an element of that order.Let |G|=8. Show that G must have and element of order 2.