Symmetry Groups and Group Actions

[Obraz35]

Homework Statement

I would like to find the number of distinct elements in S₁₇ that are made up of two 4-cycles and three 3-cycles.

Homework Equations

The Attempt at a Solution

This seems like a very simple question but since the group is so huge it's hard to figure out. I have been trying to look at the problem from the point of view of group actions but that has not gotten me anywhere yet.

 

[Dick]

Isn't it just a combinatorics question? The symmetric groups are very, uh, symmetric. How many ways to pick two four cycles and one three cycle?

 

[Obraz35]

Well, I tried looking at it that way, but it seems really complicated.
But here is what I have come up with, please let me know if I am missing something since I haven't done this type of thing in a while.

17 choose 4 = 2380
13 choose 4 = 715
9 choose 3 = 84
6 choose 3 = 20
3 choose 3 = 1

then within in each set of 4 elements there are 6 distinct permutations and within in each set of 3 there are 2 distinct permutations.

So to come up with the total would you just multiply
(2380)(715)(84)(20)(1)(6²)(2³)?

Thanks.

 

[Dick]

I thought it should be just C(17,4)*C(13,4)*C(9,3). Shouldn't the rest be fixed? So you don't have any more cycles? I think you still have to account for permutations in those cycles though.

 

[Obraz35]

Would the last two 3-cycles really be fixed? If there were six elements remaining to put into cycles there are still 20 different ways to put them into two separate 3-cycles, correct?

 

[Dick]

Would the last two 3-cycles really be fixed? If there were six elements remaining to put into cycles there are still 20 different ways to put them into two separate 3-cycles, correct?

Sure. I was reading the problem wrong. But you still have to do some more counting, don't you? Once you've chosen the set, say {1,2,3}, (123) and (132) are still different 3 cycles.

 

[Kalinka35]

Well, here is a different way of thinking about so I am not sure which way is correct..

There are 17! ways to line up the 17 elements in a row. Then we group the first 4 in a 4-cycle, the second 4 in a 4-cycle, the next 3 into a 3-cycle and so on until all the elements have been grouped together.

But there are multiple ways to get get a single element:
Switch the 2 4-cycles: 2! ways
Shift the elements inside a 4-cycle while retaining the same permutation: 4 ways
Switch the 3 cycles: 3!
Shift the elements inside a 4-cycle while retaining the same permutation: 3 ways
So this means that for a particular element of S17 there are 2!(4²)(3!)(3²)

So our final count for the number of distinct elements with this structure is
17!/[(2!)(4²)(3!)(3²)]

Does this seem right? It seems like the logic is correct but it gives me a different answer than my other method.

 

[Dick]

That sounds right. Unless I'm missing something too.

 

[latentcorpse]

$\binom {17}{4} 3! \binom {13}{4} 3! \binom {9}{3} 2! \binom {6}{3} 2! \binom {3}{3} 2!$ as there are $(k-1)!$ ways to cycle k elements

then i think you have to divide by 2 to take into account the repeated 4-cycles and then divide by 3 because of the repeated 3 cycles.