Product of 3 non-disjoint 4-cycles from $S_4$

Question

I'm having a lot of difficulty finding the products of permutations and I can't sort out why. If someone could explain to me why $$(1 2 3 4) (1 2 4 3) (1 2 4 3) = (1 3)$$ That would be great.

Answer 1

There are two conventions you need to pick, and both of them affect the answer. The first convention is whether you read compositions in the standard order (which is right-to-left, in the sense that the rightmost permutation "happens first") or the other order.

The second convention is whether permutations act on "numbers" or on "positions." For example, suppose we're trying to find the composite $(12) (23)$. We apply the permutation $(23)$ first, so we've sent $2$ to $3$ and $3$ to $2$. Now what is the result of applying $(12)$ to this? Does this mean that we swap $1$ the number and $2$ the number, or does it mean we swap the first and the second numbers? With the first interpretation the composite becomes $(123)$, and with the second interpretation the composite becomes $(132)$.

The standard convention is that permutations act on numbers. (This convention, combined with the previous convention, is the convention for which permutations are functions $\{ 1, 2, \dots n \} \to \{ 1, 2, \dots n \}$ and composition of permutations is composition of functions in the usual sense and the usual order.) So here's how to work out this composition in a way that makes it easy to check your work and compare conventions with someone else. First, write down the numbers $1, 2, 3, 4$. Next, apply each permutation, from right to left. Applying $(1243)$ gives

$$2, 4, 1, 3.$$

Applying $(1243)$ again gives

$$4, 3, 2, 1.$$

Finally, applying $(1234)$ gives

$$1, 4, 3, 2.$$

So I get that this product is $(24)$. I think you get $(13)$ from going left-to-right.