Let $G$ be a group and $a,b,c \in G$. Given that $abc$ and $cba$ are conjugated, prove that $G$ is abelian.

Question

Let $G$ be a group and $a,b,c \in G$. Given that $abc$ and $cba$ are conjugated, prove that $G$ is abelian.

In other words, if for any $a,b,c \in G$ there is a $g \in G$ so that $a b c = g c b a g^{-1}$, prove $G$ is abelian.

Answer 1

Take $c=(ba)^{-1}$. We have:

$ab(ba)^{-1}=g(ba)^{-1}bag^{-1}=ga^{-1}b^{-1}bag^{-1}=ga^{-1}1ag^{-1}=ga^{-1}ag^{-1}=gg^{-1}=1$ for $g \in G$ so:

$ab=ba$ for all $a,b \in G$