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Suppose that $G$ is a group s.t. $x^2=1$ for all $x \in G$. Prove that $G$ is abelian.

If we assume that $x^2 = 1$ for all $ x \in G$, and suppose that $a,b \in G$ and if $x=ab$ we see that $x^2 = abab =aabb = a^2 b^2$. Simplifying we see that $ab=ba$, and because we know that $x= ab$, we see that $1=1$. Therefore it is abelian. $\square$

Is this sound? I'm struggling with this stuff because I think it's so simple and I'm concerned I'm overthinking it.

I'd like to contest the mods decision that this is an exact duplicate. Though there are similar questions, as has been linked, it is not the same thing. If it is a problem with the abstract algebra tag, then delete it and allow it to be an active thread under other tags.

Bad at algebra and proofs
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    Yes, the proof is sound because of $abab=aabb=e$ – Peter Feb 23 '18 at 21:36
  • Why is it that $abab = aabb$? – Morgan Sherman Feb 23 '18 at 21:42
  • @MorganSherman: Because $x^2=1=a^2b^2$ by hypothesis. – Shaun Feb 23 '18 at 21:46
  • @Shaun, sorry, yes I know that, I meant my comment to be directed at the OP since it is not clear to me that this is clearly justified in the flow of the argument. If I were grading this proof I wouldn't give it credit. – Morgan Sherman Feb 23 '18 at 21:48
  • @MorganSherman OP is using the fact that $ayb=axb$ implies $x=y$. In this case, $y=ba$ and $x=ab$. – John Wayland Bales Feb 23 '18 at 21:50
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    @JohnWaylandBales My apologies, I don't think you get my objection. The OP has the chain of equalities "$x^2 = abab = aabb = \ldots$". My question is why is the second equality true? (Note: I know of course that it is true, e.g. by Shaun's reasoning, but I don't see how it is self-evident given the assumptions made at that point.) – Morgan Sherman Feb 23 '18 at 22:07
  • Since this just came through the review queue, where I elected to leave the question closed, let me add my two cents: you are trying to prove that $x^2=1 \forall x\in G$ implies that $G$ is abelian. This is exactly what is being asked in the previous two questions. The tagging is appropriate, and has nothing to do with why your question was marked as a duplicate. This question has several good answers. If none of those answer your question, you need to more explicitly detail what you don't understand. – Xander Henderson Feb 23 '18 at 22:25

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