# Thread: Subgroup of Abelian group

1. ## Subgroup of Abelian group

Prove that an Abelian group with two elements of order 2 must have a subgroup of order 4.

My proof:

Suppose that G is an Abelian group, and let $a,b \in G$ such that $ = 2, = 2$, so we have $a^{2}=e, b^{2}=e$.

From a theroem, we know that <a> and <b> are subgroups of G.

Am I starting this right?

We know that $G$ has two elements $a\mbox{ and }b$ which has orders two. This means $a^2 = b^2 = 1$. Now consider the elements: $1,a,b,ab$. Are these elements distinct? Well, $a\not =1 \mbox{ and }b\not =1$ (why not?). And $a\not = ab \mbox{ and }b\not = ab$ (why not?). And finally can $ab=1$? It turns out that no (why not?). If you can show that all these elements are distinct then form the subset $H= \{ 1 , a , b , ab\}$. Show that this set is a group. And hence a subgroup of order 4.