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Thread: Subgroup of Abelian group

  1. #1
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    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 $\displaystyle a,b \in G$ such that $\displaystyle <a> = 2, <b> = 2$, so we have $\displaystyle a^{2}=e, b^{2}=e$.

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

    Am I starting this right?
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Am I starting this right?
    Try this.

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