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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] $$(i)\;\;\; (1,2,3).(2,4)(1,3,5) = (1,4,2,5)$$ $$(ii)\;\;\; (1,3,4)(2,5).(2,3,4)(1,5) = (1,4,5,3,2) $$

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up vote 4 down vote accepted

Take for example the first one (from left to right)

1 goes to 2, then 2 goes to 4 and that's all, so $\,1\to 4\,$

2 goes to 3, 3 goes to 5 so $\,2\to 5\,$

3 goes to 1, 1 goes to 3. so 3 is a fixed point

4 goes to 2, so $\,4\to 2\,$

5 goes to 1 so $\,5\to 1\,$ and we cflose the cycle: $\,(1\;4\;2\;5)$

Final aclaration: it seems to me that the big majority of authors prefer to do the cycles product from right to left. Watch this when trying to read other books and, anyway, always read the author's definition of product of cycles.

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Who cares? we follow a single cycle: $$1\to 2\to 4\Longrightarrow 1\to 4\,\,\text{in this cycle's product}$$ – DonAntonio Jan 2 '13 at 16:05
@DonAntonio that's a very clear and helpful method, thank you! – Mathlete Jan 2 '13 at 16:07
Anytime, @Mathlete .... – DonAntonio Jan 2 '13 at 16:08

$$(i)\;\;\; (1,2,3)\cdot(2,4)(1,3,5) =(1,2,3)(4)(5)\cdot(1,3,5)(2,4)=(1,4,2,5)(3)=(1,4,2,5)$$ $$(1\to2)(2\to 4)=(1\to4)$$ $$(2\to3)(3\to5)=(2\to5)$$ $$(3\to1)(1\to3)=(3\to3)$$ $$(4\to4)(4\to2)=(4\to2)$$ $$(3\to5)(5\to1)=(3\to1)$$ $$1\to4\to2\to5,3\to3$$ $$(ii)\;\;\; (1,3,4)(2,5)\cdot(2,3,4)(1,5) = (1,4,5,3,2) $$ $$(1\to3\to4)\Rightarrow(1\to4)$$ $$(2\to5\to1)\Rightarrow(2\to1)$$ $$(3\to4\to2)\Rightarrow(3\to2)$$ $$(4\to1\to5)\Rightarrow(4\to5)$$ $$(5\to2\to3)\Rightarrow(5\to3)$$ $$1\to4\to5\to3\to2$$

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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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