Let $a$, $b$ and $c$ be elements of a group $G$, how can I prove that $abc$ and $cba$ do not necessarily have the same order?
I know that this cannot hold for abelian groups, but unsure how to start otherwise.
Also, it is impossible to find a counterexample if we change the order to $cab$, see here: Let $G$ be a group. Show that $\forall a, b, c \in G$, the elements $abc, bca, cab$ have the same order..
Consider the quaternion group, $$ H_{8} = \langle \pm 1,\pm i,\pm j, \pm k \mid i^2=j^2=k^2=ijk=-1 \rangle. $$ Then $ ijk = -1 $ has order $2$ (obviously, since $(-1)^2 = 1$), but $jik = -ijk = 1 $ has order $1$.
(To see this, note that $ij=(ijk)(-k)=k$, but $ji = -ji(ijk) = -k = -ji $.)