How many conjugacy classes does a group have?
Table of Contents
How many conjugacy classes does a group have?
one conjugacy class
Theorem 3.4 says each element of a group belongs to just one conjugacy class. We call an element of a conjugacy class a representative of that class. A conjugacy class consists of all xgx-1 for fixed g and varying x.
What is the order of a conjugacy class?
Theorem: The order of a conjugacy class of some element is equal to the index of the centralizer of that element. In symbols we say: |Cl(a)| = [G : CG(a)] Proof: Since [G : CG(a)] is the number of left cosets of CG(a), we want to define a 1-1, onto map between elements in Cl(a) and left cosets of CG(a).
What is the conjugate of a group?
Two elements a and b of G are called conjugate if there exists an element g in G with g∗a∗g−1=b. Here ∗ is operation on group. hence a=b.
How do you count conjugacy classes?
- The number of conjugacy classes in a finite group equals the number of equivalence classes of irreducible representations.
- The number of conjugacy classes is the product of the order of the group and the commuting fraction of the group, which is the probability that two elements commute.
Does the size of a conjugacy class divides order of group?
The number of conjugacy classes is the number of cosets of the centralizer, which is the same as the index of the centralizer. This is a divisor of the order of the group because it is the order of the group divided by the order of the centralizer.
What are the conjugacy classes of an Abelian group?
For an abelian group, each conjugacy class is a set containing one element (singleton set). Functions that are constant for members of the same conjugacy class are called class functions.
What are the conjugacy classes in S5?
Comprehensive treatment of small degrees
| Degree | Symmetric group | List of conjugacy class sizes |
|---|---|---|
| 4 | symmetric group:S4 | 1,3,6,6,8 |
| 5 | symmetric group:S5 | 1,10,15,20,20,24,30 |
| 6 | symmetric group:S6 | 1,15,15,40,40,45,90,90,120,120,144 |
| 7 | symmetric group:S7 | 1,21,70,105,105,210,210,280, 420,420,504,504,630,720,840 |
How many conjugacy classes are there in S3?
Summary
| Item | Value |
|---|---|
| conjugacy class sizes | 1,2,3 maximum: 3, number of conjugacy classes: 3, lcm: 6 |
| number of conjugacy classes | 3 See element structure of symmetric group:S3#Number of conjugacy classes |
| order statistics | 1 of order 1, 3 of order 2, 2 of order 3 maximum: 3, lcm (exponent of the whole group): 6 |