# Math Help - Lagrange Theorem.

1. ## Lagrange Theorem.

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.

2. Originally Posted by Juancd08
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.
Hint: Pair $x$ and $y$ so that $xy=e$. Note why you need the fact that $|G|$ is odd.

Let |G|=8. Show that G must have and element of order 2.
Let $a\in G - \{e \}$ and construct $\left< a\right>$. By Lagrange's theorem $|\left< a \right>| = 2,4,8$. Since this is a cyclic group and 2 divides its order we can find an element of that order.