TITLE: multiplying permutation cycles
QUESTION [1 upvotes]: I have looked up several things on here and otherwise online.  I can't understand any of the examples...Or, if I do, it seems to contradict other problems later on.
My professor wrote that (12)(123)=(23)
But I don't know why he doesn't include the 1? Everything I have seen so far makes me conclude it should be (12)(123)=(13)(2)=(13) if going from left-to-right, because 1 goes to 2 and then 3 - so, (13) - and then 2 goes to 1 which goes back to 2.  
Or, I know that it's also fine to go right-to-left, but in that case I get (1)(32)
since 1 goes to 2 and then back to one (hence the 1-cycle) and then 2 goes to 3 which stays at 3 since it doesn't appear in the first cycle)
In any case, I am not getting the same solution as my professor. Please help? Everything I look up in a textbook or online seems different.

REPLY [1 votes]: First of all, it's right-to-left, so your second computation is correct (and your first one isn't). And then, your answer $(1)(32)$ is exactly the same as your professor's $(23)$. When writing permutations in cycle notation, trivial cycles of length one are omitted — that's why there's no $1$. And $(23)$ is the same as $(32)$.
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