The product of disjoint cycles

Question

Express as the product of disjoint cycles:

$(1\ \ 2)(1\ \ 2\ \ 3)(1\ \ 2)$

It's easy to verify that above product is equal to $(1\ \ 3\ \ 2)$. Could we count the cycle $(1\ \ 3\ \ 2)$ as the disjoint? However, it sounds quite pointless.

Answer 1

No, it is not pointless at all. It's like saying that every natural number greater than $1$ can be expressed as a product of primes. What do you get if the number is $5$. You get $5$, that's all.

And here, when we talk about product of disjoint cycles, if you have a single cycle then the cycles are trivially disjoint (since there's only one of them).

Answer 2

Yes, the definition of disjoint is that it uses each number at most once.

Also, it is far from pointless: expressing a permutation in terms of disjoint cycles is essentially unique (up to reordering) whereas there are many ways of writing it as any product of cycles. This means we can define the cycle type of a permutation (e.g. the type of $(123)$ is $\{3\}$ and the type of $(12)(34)$ is $\{2,2\}$) and things which follow from there. For example, two elements of $S_n$ are conjugate if and only if they have the same cycle type.