compositions of permutations For compositions of permutations on a set $X = \{1,2,3\}$, my lecture notes say that the composition $\phi_2 \phi_1$ is the permutation $\phi_1$ followed by the permutation $\phi_2$. So consider the who permutations $$\phi_1 = (1,2,3)$$ $$\phi_2 = (1,2).$$ Splitting the permutation up gives $\phi_1 = (1,2)(2,3)$ and so the permutation $\phi_1\phi_2 = (1,2)(1,2)(2,3) = (2,3)$ but my lecture notes say that this permutation equals $(1,3)$ and the permutation $\phi_2\phi_1 = (2,3)$. What have I done wrong?
 A: You merely switched the order by mistake. You wrote $\phi_1\phi_2 = (1,2)(1,2)(2,3)$, but using the expressions for $\phi_1$ and $\phi_2$ that you gave above, this is in fact $\phi_2\phi_1$, and $\phi_1\phi_2$ should be $(1,2)(2,3)(1,2)$.
Perhaps you're misinterpreting "$\phi_2\phi_1$ is the permutation $\phi_1$ followed by the permutation $\phi_2$". That doesn't mean "followed by" in a textual sense, but "followed by" in the sense of the order of applying operations.
A: To follow up on Joriki's answer, I want to say that this is a place where the order of application of functions matters, and place the domain is written matters. If you think of three dots at the top of a page labeled 1,2,3 in order, when you apply (12) below the dots, obviously the first and second dots have been interchanged. You can write that as (X|). Now the permutation (123) should be written in diagrammatic or string notation as (|X)(X|) That is 2 switches with 3, and then 1 switches with 2. So the composition is written (X|)(|X)(X|). Here the domain of the composition is on the right, and the order of composition is domain $\phi_2$, $\phi_1$. 
I found the situation highly confusing (and I still do when reading other authors). What you have to do is develop a convention that you will use for the rest of your life, and make sure that your convention gives what your lecturer wants. To be sure, write out each of the 24 permutations of ${1,2,3,4}$, try a diagrammatic schemata, and check with your instructor that you have the correct convention. 
