Question

Prove that every 4-cycle can be written as a product of 2-cycles
Please, help

Answer 1

@rational @myininaya

Answer 2

(1,2,3,4,1) = (...)(...)

something like this ? if possible, can you teach me quick the definition of product in regards to groups ?

Answer 3

I had to look this up.
http://www.math.niu.edu/~jthunder/Courses/2009Fall/420/sec1/oct12/permutations2.pdf
The site seems wise. I'm reading now.

Answer 4

ehh.... In my book, it says (1,3,2,4) can be written as (13)(12)(14)

Answer 5

all I have to do is prove (1324) = (13)(12)(14) but I can't get the right answer.

Answer 6

@rational I found this link to be really helpful with definitions: http://people.math.sfu.ca/~jtmulhol/math302/notes/4-Permutations-CycleForm.pdf

Answer 7

\(\left[\begin{matrix}1&2&3&4\\3&4&2&1\end{matrix}\right]\) =\(\left[\begin{matrix}1&2&3&4\\3&2&1&4\end{matrix}\right]\)\(\left[\begin{matrix}1&2&3&4\\2&1&3&4\end{matrix}\right]\)\(\left[\begin{matrix}1&2&3&4\\4&2&3&1\end{matrix}\right]\)

Answer 8

Exactly what I am looking for! thank you so much

Answer 9

That 's what I have, but I can't get them equal.

Answer 10

Start from far right, the last 2 terms gives me \(\left[\begin{matrix}1&2&3&4\\4&1&3&2\end{matrix}\right]\)

Answer 11

(1,2,3,4) = (1,2,3)(1,3,4)

Answer 12

\[(1,2)(1,3)(1,4)\]
\[1\xrightarrow{(1,3)}3\xrightarrow{(1,2)}3\xrightarrow{(1,4)}3\\
3\xrightarrow{(1,3)}1\xrightarrow{(1,2)}2\xrightarrow{(1,4)}2\\
2\xrightarrow{(1,3)}2\xrightarrow{(1,2)}1\xrightarrow{(1,4)}4\\
4\xrightarrow{(1,3)}4\xrightarrow{(1,2)}4\xrightarrow{(1,4)}1\]
This gives a cycle \((1,3,2,4)\), so indeed they are equivalent.

Answer 13

combine with the first one from the right, I have \(\left[\begin{matrix}1&2&3&4\\4&3&1&2\end{matrix}\right]\) which is not the left hand side
@rational it's the product of 3-cycle

Answer 14

Oh the proof is about writing 4-cycle as products of 2-cycles... I see thanks :)

Answer 15

@SithsAndGiggles It should give the same answer when I do it under the matrix form, right? but they are not, why? It frustrates me a lot. I got confuse.

Answer 16

I understand the way you do. I like it too. But I have to follow my prof's way (matrix way)

Answer 17

It should work with whatever notation you choose. (I'd just learned this stuff a little less than an hour ago actually, so I just went with the most economic notation :) )

Answer 18

Please, check my work and point out where my correct is.

Answer 19

If your books says (1324)=(13)(12)(14), then that is incorrect. If we are using the standard idea of product and going from right to left, then (1324)=(14)(12)(13)

Answer 20

*not hihihi

Answer 21

@nerdguy2535 give me some link, please

Answer 22

Sure, one sec.

Answer 23

One more thing, you see @SithsAndGiggles proof above showed that (13)(12)(14) works well. And we know that non-disjoint cycles is not commutative.
--> only 1 method is a correct one. Yours or Sith??

Answer 24

I was under the impression left-to-right was the standard method...

Answer 25

I think we are reading this: (13)(12)(14) differently. So before we proceed lets verify something. When I see (132)(24), and someone asked me where 4 went to, I would say, "4 goes to 2, and then 2 goes to 1, so 4 goes to 1". Is that how you would read this?

Answer 26

yes

Answer 27

and it is a multiplication of 2 cycles, not a product of a non-disjoint cycle, right?

Answer 28

Alright, then I stand by my statement that (1324) is not the same as (13)(12)(14), since 4 goes to 1 in the 4 cycle, and 4 goes to 2 in the product of 2 cycles. This is why your matrix multiplication isnt coming out to the right answer. You should be doing (14)(12)(13) instead.

Answer 29

I'd read it as \(4\to4\), then \(4\to2\).

Answer 30

@nerdguy2535 I got you
@SithsAndGiggles where is your question from?

Answer 31

Then for you, (13)(12)(14) is correct. and you would multiply the matrices from left to right and get the right answer.

Answer 32

Okay so we agree that we get two different results using two fundamentally different methods... My work suggests @Loser66 is supposed to use the left-to-right interpretation. Is that how your text/professor is teaching this?

Answer 33

@nerdguy2535 not just that, please, give me link, I don't want to confuse anymore. I have to master the concept.

Answer 34

Here's a link to an example
http://en.wikipedia.org/wiki/Symmetric_group#Multiplication

Answer 35

@SithsAndGiggles my prof teaches me definition, theorem, proof of theorem, and... done.
I search on internet and study myself to do homework.

Answer 36

@nerdguy2535 I read it, it doesn't talk about decompose from left to right or right to left

Answer 37

In their example they compute (13)(45)(125)(34) and get (124)(35). You could only get this if you went right to left.

Answer 38

I realize that some mathematicians use right to left, some use left to right. I read and confused.

Answer 39

Also, I don't think we are getting two different results, I just think we are accidentally mixing up the two ways of reading these things >.<

Answer 40

Like, if we had known we were taught to read these differently, there wouldnt have been a problem.

Answer 41

Ok, I would like to make sure again
if the cycle is (12435) , then it is = (15)(13)(14)(12), right?

Answer 42

Yeah. The link that myininaya posted has the general formula for decomposing a k-cycle into two cycles.

Answer 43

Keep in mind that answer is not unique though. There are infinitely many ways to decompose a given cycle into two cycles.

Answer 44

what if it misses some, like (145) , it is = (15)(14) , right?
and I can rearrange (145) as (451) , so that it is (41)(45) , right?

Answer 45

Yes, that all works.

Answer 46

Thanks a ton. :)

Answer 47

@nerdguy2535 this is the link I see them decompose the product from left to right. They don't give out the procedure so that I don't know whether they proceed it from left to right or from right to left. And it confused me a lot
I mean: On page 3, part Decomposition of k-cycles into 2cycles, they give example \(\alpha = (1, 3, 5)(2, 4, 7,6,8) = (1, 3)(1,5)(2, 4)(2, 7)(2, 6)(2, 8)\)
Obviously, from the right hand side, they break it from left to right, right?
but when I applied the rule right to left, I got wrong!!! Ha!!!