When to close brackets in product of disjoint cycle

Question

when expressing 2 composition of function as a product of disjoint cycles, when do we 'close' the bracket?

None of the sources explain this clearly. Some do not even make an attempt to.

Answer 1

I am making a guess about your question. This is about how to present a permutation, rather than about composition of function.

Standard way of presenting a permutations is listing out the values of the function at 1, at 2, at 3 and so on in that order (remember permutation is a bijective function to the same set).

In disjoint cycle form we present it in an alternative format:

First write the value f the function at 1. Then we take this value, and then list the value of this function there. That is $f(1), f(f(1)),$ and then $f\big( f(1))\big)$. When we do this we may come back to the initial value 1. Instead of writing this repetion we close the bracket at previous value.

Now this is just one one of the component cycles. Then we take a number that did not get listed and open a bracket with that number as the initial value for the next cycle, and keep applying $f$ and list them as above using the same rule, closing it just before the repetition of the initial value of the second bracket.