Is it always true that $(a,b,c)(a,b,c) = (a,c,b)$? I noticed that $(1,2,3)(1,2,3) = (1,3,2)$, and I also noticed that $(1,4,3)(1,4,3) = (1,3,4)$. Now, my question is whether or not it is true that for any permutation $(a,b,c)^2 = (a,c,b)$?

Here's a purely computational answer. Since a permutation of a set is uniquely determined by its action on that set, let us consider how $(a,b,c)^2$ acts on $S$. Take $s \in S$. If $s$ is not one of $a,b,c$, then $(a,b,c)^2$ fixes $s$. Otherwise, we compute
$$a \mapsto b\mapsto c$$
$$b \mapsto c \mapsto a$$
$$c \mapsto a \mapsto b$$
so $(a,b,c)^2 = (a,c,b)$.
