I just want to address the idea of disjoint cycles and product of cycles because I believe I am confusing the idea.
My professor said:
Write $(2\:5)(6\:4\:7)(2\:4\:5\:3)$ as a product of disjoint cycles in $S_7$
where the answer is: $$(2\:5)(6\:4\:7)(2\:4\:5\:3)=(2\:7\:6\:4)(3\:5)$$
but the idea here is he moved from right to left.
Now I'm working out problems and I encountered this problem:
Consider the permutation $a=(1\:4\:6\:3\:7)(2\:8)$ and $b=(2\:5\:3)(4\:7\:8\:1)$ in $S_8$
but in order to calculate the product, the work was done from left to right?
So my question is why for disjoint cycles, you move from right to left but for products left to right?