If at least one pair of the elements $a,b,c \in G$ commute, then $G$ is abelian.

Question

Let $G$ be a group and consider three arbitrary elements $a,b,c$ in $G$. If at least one pair of these elements commute (i.e. at least "$ab=ba$" or "$ac=ca$" or "$bc=bc$" is valid), then $G$ is an abelian group.

Answer 1

$(u,v,uv)$ $u$ commutes with $v$ done. Else we have $u$ commutes with $uv$, $u(uv)=(uv)u$ by multiplying the left side of the last equality by by $u^{-1}$, $uv=vu$.