Conjugates in symmetric groups

In summary, the question is asking how many conjugates the elements (1,2,3,4) and (1,2,3) have in the symmetric groups S7 and S5, respectively. The answer for (1,2,3) in S5 is 10, found using the binomial coefficient formula. However, the attempt at finding the answer for (1,2,3,4) in S7 by manually listing the conjugates was unsuccessful. The person asking the question wonders if there is a statistical or mathematical equation that can be used to find the number of conjugates in this case. It is suggested that the binomial coefficient formula may be relevant, but upon further consideration, it is realized that
  • #1
kimberu
18
0

Homework Statement


The question is, "How many conjugates does (1,2,3,4) have in S7?

Another similar one -- how many does (1,2,3) have in S5?


The Attempt at a Solution


I know that the conjugates are all the elements with the same cycle structure, so for (123) I found there are 20 conjugates by hand listing them (132)...(354) etc. But I was wondering if there's some equation to find this amount- I haven't taken probability but I think there's got to be a statistical way to figure it out! Thanks a lot.
 
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  • #2
the binomial coefficient might be what you're looking for.
"nCr"
"choose function"
 
  • #3
using the formula on (1,2,3) in S5, I got that it has 10 conjugates, which is wrong -- which N and R should I use for this example if not 5 and 3 (or am I calculating wrong)?
 
  • #4
No, you're doing it right (i.e. "this isn't what you're looking for). I forgot to divide by 2! (!)... :/
 

1. What is a conjugate in a symmetric group?

A conjugate in a symmetric group is a permutation that can be obtained by rearranging the elements of another permutation in a specific pattern. In other words, two permutations are conjugates if they have the same cycle structure.

2. How do you determine if two elements in a symmetric group are conjugates?

To determine if two elements in a symmetric group are conjugates, you can compare their cycle structures. If the cycle structure of one permutation can be obtained by rearranging the elements of the other, then they are conjugates.

3. Are all elements in a symmetric group conjugates of each other?

No, not all elements in a symmetric group are conjugates of each other. Two elements are only conjugates if they have the same cycle structure.

4. What is the significance of conjugates in symmetric groups?

The concept of conjugates in symmetric groups is important in understanding the structure and properties of symmetric groups. It allows for the classification and organization of elements in a symmetric group based on their similarity in cycle structure.

5. Can a symmetric group have more than one conjugacy class?

Yes, a symmetric group can have multiple conjugacy classes. The number of conjugacy classes in a symmetric group is equal to the number of distinct cycle structures, which can vary depending on the size of the group.

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