Q. Find the product of $\alpha \beta $ where they are permutations and equal:
$$\alpha=(1,5,2,3)$$ and $$\beta=(1,5,4)(2,3)$$
Then the product is:
$$\alpha \beta = (1,4)(3,5)$$ but when I worked it out I got:
$$\alpha \beta =(1,5,2,3)(1,5,4)(2,3)=(1,2)(4,5)$$
so where did I go wrong?
There is no agreement in the literature about whether permutations should be composed right to left or left to right. Both conventions are used. Most authors will explicitly say which convention they are using. (I always look for that when I am reading about permutations.)
The answer given is based on left-to-right composition: $(1,5,2,3)(1,5,4)(2,3) = (1,4)(3,5)$ by following indices left-to-right through the permutations: $1\mapsto 5\mapsto 4$, and $4\mapsto 1$, so $(1,4)$ is in the product. And then $2\mapsto 3\mapsto 2$, so $2$ is fixed. And then $3\mapsto 1\mapsto 5$, and $5\mapsto 2\mapsto 3$, so $(3,5)$ is in the product.
Your answer is close to correct with the right-to-left composition convention, but you made a slight composition error. With right-to-left composition, we have $1\mapsto 5\mapsto 2$, and $2\mapsto 3\mapsto 1$, so $(1,2)$ is in the product; and $3\mapsto 2\mapsto 3$, so $3$ is fixed, and $4\mapsto 1\mapsto 5$ while $5\mapsto 4$, so $(4,5)$ is in the product. Thus with this convention the right answer is $(1,2)(4,5)$.
[EDIT: I see you have fixed your calculation. Your answer is correct with the right-to-left convention.]
