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.

Question

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.

Answer 1

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 $b$?

Answer 2

$H=\{e,a,b,ab\}$ is a subgroup of $G$. By Lagrange's Theorem, $|H|$ divides $|G|$