Multiplying cycles

Question

How does one perform cycle multiplication? It seems that every textbook I read has a different notation for this and it's not clear at all. Suppose I have $(123)(134)$ now some book stated that the multiplication should go from right-to-left, but I couldn't find what to do with for example $4$ here since that doesn't map to anything? Doesn't that break the whole thing since it's not a bijection?

Answer 1

When we multiply the two permutations $(123)$ and $(134)$ in that order, that (usually) means we first apply $(134)$, and then apply $(123)$ to the result:

• $1$ is first sent to $3$ by $(134)$. Then that $3$ is sent to $1$ by $(123)$
• $2$ is first sent to $2$ by $(134)$. Then that $2$ is sent to $3$ by $(123)$
• $3$ is first sent to $4$ by $(134)$. Then that $4$ is sent to $4$ by $(123)$
• $4$ is first sent to $1$ by $(134)$. Then that $1$ is sent to $2$ by $(123)$

Thus we see that the end result of applying this product is the same as for the permutation $(234)$. Therefore we have $(123)(134) = (234)$.