How do I do cycle notation?

Question

Consider the three permutations given in cycle form: $f = (124)(35)$, $g = (13)(45)(2)$, $h = (1)(4253)$ Which composition of the functions $(f, g ,h)$ gives the permutation $(1)(2)(34)(5)$?

Answer 1

Note that $\sigma = (123)(4)(5)$ is just another notation for the map $$\sigma\colon\{1,2,3,4,5\}\to\{1,2,3,4,5\}\\ 1\mapsto 2\\ 2\mapsto 3\\ 3\mapsto 1\\ 4\mapsto 4\\ 5\mapsto 5 $$. So composition is nothing else than composition of maps. That is, given $\tau = (12345)$, you have $(123)(4)(5)(12345)= \sigma\circ\tau$. To evaluate this, you have to evaluate the maps and look where every element is mapped to. In particular, you have to evaluate from right to left, since the right map is applied first.

For example, if you want to evaluate $f\circ g$ in your exercise, first write down the composition of cycles: $$ f\circ g = (123)(35)(1)(4253)$$ and then look where every element's mapped to: evaluating from right to left and chosing $1$ as our starting point, it first appears in the 3rd cycle, which yields $1\mapsto 1$. Now you have to check, where the target is going. Which in this case this is again $1$, which (evaluating from right to left, starting at the second cycle, since you've already used the 4th and the 3rd) next appears in the first cycle, where it's getting mapped to $1$, so $1\mapsto 1\mapsto 2$. So you can write down: $$f\circ g = (12\dotsc$$ So you have to check, where the $2$ is going. It appears first in the 4th first cycle, where it's getting mapped to 5, so $2\mapsto 5$. So you have to check where $5$ is getting mapped to. Evaluating from right to left, starting at the 3rd cycle, it first appears in the 2nd cycle, where it's mapped to 3, so $2\mapsto 5\mapsto 3$, looking for the three and finding it in the first cycle, you'll get $2\mapsto 5\mapsto 3\mapsto 1$, hence $$f\circ g = (12)\dotsb$$ Repeating these steps, you should get $$3\mapsto 4\ \implies f\circ g = (12)(34\dotsc\\ 4\mapsto 2 \to 3 \implies f\circ g = (12)(34)\dotsb\\ 5\mapsto 3\to 5 \implies f\circ g = (12)(34)(5)$$ Evaluate all possible combinations of $f$, $g$, and $h$, and you will find the permutation your looking for.