I'd be grateful for some help reading permutation symbols such as $(123)$. Does it mean, when applied to a target sequence such as $(x y z w)$, "replace the element in the first slot of the target with the element in the second slotof the target, the element in the second slot of the target with the element in the third slot of the target, and the element in the third slot of the target with the element in the first slot of the target," resulting in $(y z x w)$? If so, applied twice, $(123)$ would produce $(z x y w)$, which would mean that $(123).(123)=(321)$?
If I'm getting it right, then I should imagine little leftward arrows inside the permutation symbol, as in $(\leftarrow 1 \leftarrow 2 \leftarrow 3)$, with the first arrow implicitly wrapping around to the last position of the permutation symbol.