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Let $G$ be a finite group. Let $a,b\in G$ be two distinct elements or order 2. Prove that if $ab=ba$, then the order of $G$ is divisible by 4.

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HINT: To prove that the order of $G$ is divisible by $4$, you simply need to show some subgroup of order $4$ exists (by Lagrange's theorem). Can you create such subgroup with what you know of $a$ and $b$?

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  • $\begingroup$ I have tried to find an element whose square is a or b. But there's no guarantee such element exists in G. $\endgroup$ – shansh0201 Nov 10 '16 at 9:41
  • $\begingroup$ No, there indeed isn't. But you don't need that either. $\endgroup$ – TastyRomeo Nov 10 '16 at 9:43
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$H=\{e,a,b,ab\}$ is a subgroup of $G$. By Lagrange's Theorem, $|H|$ divides $|G|$

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