I have been asked to show that in Sn the cycle (1,2,…n) only commutes with its powers.
I know that cycles commute when they are disjoint and that every permutation can be written as a product of disjoint cycles but how do i show that this cycle and its powers are disjoint?
Could someone please help with this problem or point me to a helpful resource or book, i would greatly appreciate it!
Asked, by who?
We don’t do your homework around here, but just to get you off the wrong track: you can’t show that this cycle and its powers are disjoint cycles, because they won’t be. The power may not even be a cycle, and even on decomposing the power into a product of disjoint nonempty cycles, none of those cycles will be disjoint from the original cycle (how could they be? The original cycle contains “everything”. No nonempty cycle is disjoint from it.). Only in the specific trivial case where the power you’re looking at is the identity permutation might you be able to get away with an argument from disjointness.
So you’ll have to solve the problem another way.
Incidentally, it looks like you’re spending your effort trying to show that this cycle commutes with its powers rather than that it fails to commute with anything else. That it commutes with its powers is easy (can you see why everything commutes with its own powers?); where your effort needs to go is in showing that it fails to commute with anything else.
To expand on what Indistinguishable said, the problem–as posed–is even wrong. It must have specified that it was in S_n, since in S_{n+2} that cycle is disjoint from, and therefore commutes with, (n+1 n+2). With that proviso, the problem is quite easy.
You are definitely right!
The way i asked the question was most likely wrong.
The book I’m using for Abstract algebra offers the statement without proof stating that the proof is obvious and easy to show. But it seems to elude me!
However i looked at the book again and it did specify S_n !!
Thank you for your help.
It was pointed it out to me that the way i asked the question was wrong and made it sound like a homework thing, sorry about that.
I appreciate the help !
I was indeed going about this the wrong way, it is indeed not so hard to prove that any cycle in S_n commutes with its own powers but not with everything else is the main heart of the question. Thank you for pointing that out.
I wonder if anyone could point me to a book or a link where they would offer a proof of this statement, since the book i use does not do that!
Thanks again.
P.S. This is not a homework I’m trying to teach the course to myself.