How Can You Find the Number of Conjugation Permutations in a Group?

In summary, the number of conjugation permutations (s) in a group of permutations with 5 objects that also conjugate a and b, where a = (1 4 2)(3 5) and b = (1 2 4)(3 5), is 6. This can be calculated by finding the centralizer of a in S_5, which is equal to the number of elements in S_5 divided by the size of the conjugacy class of a, which in this case is 20.
  • #1
physicsjock
89
0
Hey,

I just have a small question regarding the conjugation of permutation groups.

Two permutations are conjugates iff they have the same cycle structure.

However the conjugation permutation, which i'll call s can be any cycle structure. (s-1 a s = b) where a, b and conjugate permutations by s

My question is, how can you find out how many conjugation permutations (s) are within a group which also conjugate a and b.

So for example (1 4 2)(3 5) conjugates to (1 2 4)(3 5) under s = (2 4), how could you find the number of alternate s's in the group of permutations with 5 objects?

Would it be like

(1 4 2) (3 5) is the same as (2 1 4) (35) which gives a different conjugation permutation,
another is

(4 1 2)(3 5), then these two with (5 3) instead of ( 3 5),

so that gives 6 different arrangements, and similarly (1 2 4) (35) has 6 different arrangements,

and each arrangement would produce a different conjugation permutation (s)

so altogether there would be 6x6=36 permutations have the property that
s-1 a s = b ?

Would each of the arrangements produce a unique conjugation permutation (s) ?
I went through about 6 and I got no overlapping conjugation permutations but I find it a little hard to a imagine there would be unique conjugation permutations for each of the 36 arrangements.

Thanks in advance
 
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  • #2
I'm really confused by your question. Every single s will produce a conjugate of a, namely ##sas^{-1}##. Of course, different s and t might give the same conjugate ##sas^{-1}=tat^{-1}##.

But surely that's not what you're asking about... Did you intend to say that you have a fixed a and b in S_5, and you want to count the number of elements s such that ##sas^{-1}=b##?
 
  • #3
Yea that's right I want to count the number of s for fixed a and b,

Sorry for not explaining it well,

Is the way I wrote correct for a = (1 4 2)(3 5) and b = (1 2 4)(3 5)?

The first s would be (2 4),

Then rewritting a as (2 1 4)(3 5) the next would be (1 2)

Then rewritting as (4 2 1) (3 5) to get another (1 4)

Then (1 4 2)(5 3) gives (3 5)

and so on

I checked and each of these, (2 4), (1 2), (1 4) and (3 5) correctly conjugate (1 4 2)(35) to b

so would that suggest there are 6 different possible s for a and b?

Since there are 3 arrangements of (1 4 2) and 2 arrangements of (3 5) which give the same permutation.

Thanks for answering =]
 
  • #4
Yes, that's correct.

You can get a formula for general a and b (of the same cycle type) in S_n as follows. Begin by noting that $$ |\{s\in S_n \mid sas^{-1}=b\}| = |\{s\in S_n \mid sas^{-1}=a\}|. $$ But the RHS is simply the order ##|C_{S_n}(a)|## of the centralizer of a in S_n, and this is the number you want. Now recall that the order of the centralizer of a is equal to the order of S_n divided by the size of the conjugacy class of a (this follows, for example, from the orbit-stabilizer formula), and there is a general formula for the latter - see e.g. here.

Let's work this out for a=(142)(35) in S_5. The size of the conjugacy class of a is (5*4*3)/3=20, so the order of the centralizer of a is 5!/20=6, confirming your answer.
 
  • #5
for your help!

Hi there,

Thank you for your question about conjugation of permutation groups. Finding the number of conjugation permutations that can also conjugate a and b can be a bit tricky, but I can provide some guidance to help you figure it out.

Firstly, let's define what a conjugation permutation is. A conjugation permutation is a permutation that, when applied to a permutation group, results in a conjugate permutation. In simpler terms, it is a way of rearranging the elements of a permutation group to create a new permutation that has the same cycle structure as the original one.

Now, let's look at the example you provided. You correctly identified that (1 4 2)(3 5) and (1 2 4)(3 5) are conjugate permutations under the conjugation permutation (2 4). However, you also found other arrangements that also result in the same conjugate permutation, such as (2 1 4)(3 5) and (4 1 2)(3 5).

To find the number of unique conjugation permutations, we need to consider the possible arrangements of the two cycles (1 4 2) and (3 5). Since each cycle has 3 elements, there are 3! = 6 possible arrangements for each cycle. Multiplying these together, we get 6x6=36 possible arrangements of the two cycles. However, not all of these arrangements will result in a unique conjugation permutation.

For example, (1 4 2)(3 5) and (2 1 4)(3 5) will result in the same conjugation permutation, since the only difference is the starting point of the first cycle. Similarly, (4 1 2)(3 5) and (1 2 4)(5 3) will also result in the same conjugation permutation.

To avoid counting these duplicates, we can divide the total number of arrangements by the number of possible starting points for each cycle, which is 3 for a 3-element cycle. So, the total number of unique conjugation permutations for (1 4 2)(3 5) and (1 2 4)(3 5) is 36/3=12.

In general, for a permutation group with n elements, the number of unique conjugation permutations will be n!/3 for
 

FAQ: How Can You Find the Number of Conjugation Permutations in a Group?

1. What is a permutation group conjugate?

A permutation group conjugate is a group element that can be obtained by rearranging the elements of another group element according to a fixed permutation. In other words, two group elements are conjugates if they are equivalent up to a relabeling of their elements.

2. How are permutation group conjugates related to group isomorphisms?

Permutation group conjugates are closely related to group isomorphisms. A group isomorphism is a bijective map between two groups that preserves the group structure. In other words, two groups are isomorphic if their elements can be relabeled in such a way that they become conjugates of each other.

3. What is the significance of permutation group conjugates in algebraic structures?

Permutation group conjugates play a crucial role in understanding the structure of algebraic objects such as groups, rings, and fields. They help to identify isomorphic structures, and can also be used to classify groups based on their conjugacy classes.

4. Can permutation group conjugates be used to simplify calculations in group theory?

Yes, permutation group conjugates can be used to simplify calculations in group theory. By relabeling elements, conjugates can often be transformed into a more manageable form. This can make it easier to prove group properties, solve equations, and perform other calculations.

5. How are permutation group conjugates related to the concept of symmetry?

In mathematics, symmetry refers to the invariance of a system under certain transformations. Permutation group conjugates are closely related to this concept, as they represent different ways of rearranging the elements of a group while preserving its structure. This allows for a deeper understanding of symmetry in various mathematical contexts.

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