Two row notation and cycle notation

Question

I am a little confused as to how we can convert from cycle notation to two row notation. say we have, $(3 1 2 4 5)(4 2 1 3)$, $3$ chooses $1$ but $2$ also chooses $1$ and the function has to be a bijection. How would I then express this in two row notation (note: I don't know if this example actually works but my point is how do I read this because in my eyes $1$ is being chosen twice).

Answer 1

Multiply out the product of cycles first (I'm doing this left to right but the convention varies).

In your example $$ (31245)(4213) $$ you work out what happens one element at a time. So $1 \to 2$ in the first cycle, then $2 \to 1$ in the second. That means $1$ is fixed by the product. Then $2 \to 4 \to 2$, $3 \to 1 \to 3$, $4 \to 5$ and does not move in the second cycle. Finally, as a check, $5 \to 3 \to 4$, as it must.

Thus $$ (31245)(4213) = (1)(2)(3)(45) $$ which is easy to write in two row notation.

You will always get disjoint cycles when you carry out the multiplication of cycles.

I hope you never need two row notation. Cycle notation tells you much more at a glance.