TITLE: Simplify these products of permutations
QUESTION [2 upvotes]: Can someone please explain how to get these answers? Everytime I think I understand the method, I end up getting a completely different answer to the one provided. How exactly do you go about simplifying these?
[Note: Read from left to right]
$$(i)\;\;\; (1,2,3).(2,4)(1,3,5) = (1,4,2,5)$$
$$(ii)\;\;\; (1,3,4)(2,5).(2,3,4)(1,5) = (1,4,5,3,2) $$

REPLY [2 votes]: $$(i)\;\;\; (1,2,3)\cdot(2,4)(1,3,5) =(1,2,3)(4)(5)\cdot(1,3,5)(2,4)=(1,4,2,5)(3)=(1,4,2,5)$$
$$(1\to2)(2\to 4)=(1\to4)$$
$$(2\to3)(3\to5)=(2\to5)$$
$$(3\to1)(1\to3)=(3\to3)$$
$$(4\to4)(4\to2)=(4\to2)$$
$$(3\to5)(5\to1)=(3\to1)$$
$$1\to4\to2\to5,3\to3$$
$$(ii)\;\;\; (1,3,4)(2,5)\cdot(2,3,4)(1,5) = (1,4,5,3,2) $$
$$(1\to3\to4)\Rightarrow(1\to4)$$
$$(2\to5\to1)\Rightarrow(2\to1)$$
$$(3\to4\to2)\Rightarrow(3\to2)$$
$$(4\to1\to5)\Rightarrow(4\to5)$$
$$(5\to2\to3)\Rightarrow(5\to3)$$
$$1\to4\to5\to3\to2$$
{"set_name": "stack_exchange", "score": 2, "question_id": 269231}