<s>[INST] Simplify these products of permutations 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]
$$(i)\;\;\; (1,2,3).(2,4)(1,3,5) = (1,4,2,5)$$
$$(ii)\;\;\; (1,3,4)(2,5).(2,3,4)(1,5) = (1,4,5,3,2) $$
 [/INST] To find, for example, $(1,2,3)\cdot(2,4)(1,3,5)$ you need to apply it on every $1\leq i\leq 5$:
$$\begin{array}{l}
[1](1,2,3)\cdot(2,4)(1,3,5)=[2](2,4)(1,3,5)=4\\
[2](1,2,3)\cdot(2,4)(1,3,5)=[3](2,4)(1,3,5)=5\\
[3](1,2,3)\cdot(2,4)(1,3,5)=[1](2,4)(1,3,5)=3\\
[4](1,2,3)\cdot(2,4)(1,3,5)=[4](2,4)(1,3,5)=2\\
[5](1,2,3)\cdot(2,4)(1,3,5)=[5](2,4)(1,3,5)=1
\end{array}$$
Hence, by writing it as a permutation, starting with one, you have: $(1,4,2,5)$
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