Can I write $(12)(34)$ as one cycle?

Question

I have trouble understanding an exercise. I wanted to know if I can write a cycle like $(12)(34)$ as one cycle instead of a multiplication?

Answer 1

No: the decomposition as a product of disjoint cycles is unique, up to the order.

For this one particularly, such a cycle would involve $1,2,3$ and $4$, hence it would be cycle of length $\ell\ge 4$, and its order would be $\ell-1\ge 3$, whereas $(12)(34)$ has order $2$.