TITLE: Proving a group is Abelian
QUESTION [9 upvotes]: Let $G$ be a group with the property that in every subset of 4 distinct elements, there exists at least a pair of commuting elements.
Show that G is Abelian.

I have thought so far that if G isn't abelian then if x,y dont commute and given subset of index 3 then the subset $\{x,y,xy\} \implies xy = yx$. Can I find something similar for the 4-element case?

REPLY [23 votes]: The theorem is incorrect. The quaternion group $Q_8$ is a counterexample, where $Q_8 = \{1, i, j, k, -1, -i, -j, -k\}$. 
In this group, every element commutes with its negative. Also, $1$ and $-1$ commute with eveything. By the pidgeonhole principle, every 4-element subset of $Q_8$ either contains $1$, $-1$, or both an element and its negative. Hence, there is a commuting pair in every 4-element subset of $Q_8$. 
But $Q_8$ is non-abelian, since $ij \ne ji$. 
Thanks to @DanielFisher for his help.