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I have to prove the statement in the title. As a lengthy hint the following is given:

Hint: let $a \in G$ $\not= e$. Suppose $a$ has period $p$ (for period $p^2$ the answer is trivial), then consider the subgroup $H = <a>$. Prove $b \notin H$ implies $G = <a, b>$. Is $H$ normal? Prove $a$ and $b$ commute and conclude $G$ is abelian.

Getting to the point where I prove that $G = <a,b>$ is no problem. However, I don't see on how to proceed to prove that $H$ is normal and assuming that, that $a$ and $b$ commute. Some help would be greatly appreciated.

penguin131415
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  • http://math.stackexchange.com/questions/212084/group-with-order-p2-must-be-abelian-how-to-prove-that duplicate? – 3SAT Jun 01 '15 at 20:03
  • Assuming you've gotten to the point you claim to have reached: in order to prove that $H$ is normal, it suffices to show $bH = Hb$ – Ben Grossmann Jun 01 '15 at 20:03

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