Product of Permutations- Abstract Algebra I'm trying to find the product of the permutations: (1,2,3)(2,4,3,5)(1,3,2) 
From what I've been taught in my Abstract Algebra class, the product of permutations is in fact a composition of permutations. So, in cyclic notation, I got my answer to be (1,5,3,4) (2). However, Wolfram Alpha tells me that my answer is incorrect. I must be missing something here. The way I solve these is the following: I treat each permutation as a function. So let's say our 3 functions are f,g,h. f composed with g composed with h = f(g(h)). So let's say I want to know where 3 goes to, I take f(g(h(3))= f(g(2))= f(4)= 4. And I do this for all of the other elements. What am I doing wrong here? Any help would be appreciated.
As it says on the W|A page, "the product $\sigma \tau$ represents the permutation $\tau(\sigma(\cdot))$." So they're just using the opposite, "left-to-right" notation. For instance, they get the cycle $(3)$ because $3$ is mapped to $1$, then $1$ is mapped to $1$, then $1$ is mapped to $3$.
this opposite way of expressing the composition is quite prevalent. Even JA Gallian does this way in his book on abstract algebra.