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 $(a,b,c)^2 = (a,c,b)$?

This is true for 3 cycles because a 3 cycle generates a cyclic subgroup of order 3, which has only the 3 cycles $(a,b,c)$ and $(a,c,b)$, and given any nonidentity element $x$ of this subgroup, $x^2$ is the other nonidentity element. In particular, $(a,c,b)^2=(a,b,c)$.
