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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.

PBJ
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  • This one helped me a lot, for anyone else who is struggling to understand this. http://math.stackexchange.com/questions/31763/multiplication-in-permutation-groups-written-in-cyclic-notation – PBJ Mar 22 '17 at 12:32

1 Answers1

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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 $(32)$.

zipirovich
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  • Oh...right. I also was misunderstanding the method a little bit too, but somewhow it makes sense now. Thanks :) (23)=(32) is good to know too. – PBJ Mar 22 '17 at 12:21
  • @PBJ: And yes, any cycle can be equivalently written in as many ways as there are elements in it. Imagine a drawing of a cycle: elements going one into another around a circle. You can list them starting from any one -- you can always come around the full circle. That's why, for example, $(1234)=(2341)=(3412)=(4123)$. It's pretty customary to start writing a cycle from the smallest element -- but it's only a fairly common tradition, not a rule at all. – zipirovich Mar 22 '17 at 12:30