Full explanation of calculating conjugate classes of the group S3
The group S3, also known as the symmetric group of order 3, consists of all possible permutations of three objects. The elements of S3 are as follows:
- The identity permutation, denoted e, which fixes all three objects.
- Two transpositions, denoted (1 2) and (2 3), which exchange pairs of objects.
- Three 3-cycles, denoted (1 2 3), (1 3 2), and (2 1 3), which cycle the objects in a cyclic order.
To calculate the conjugate classes of S3, we must first determine the conjugacy relations between its elements. Two elements g and h of a group G are conjugate if there exists an element a in G such that h = aga^-1. In other words, h is obtained by conjugating g by the element a. Conjugate elements have the same cycle structure or permutation structure. This is because conjugation by an element a permutes the objects in the same way that a acts on the group.
To determine the conjugacy classes of S3, we consider each element in turn and conjugate it by all other possible elements of the group. We use the cycle notation to represent the elements of S3. Let's begin with the identity permutation, e. Conjugating e by any element a of S3 yields the same element e. Therefore, the conjugacy class of e is {e}.
Next, we consider the transposition (1 2). Conjugating it by the identity permutation yields (1 2), which is not equal to (1 2). Conjugating it by (1 3) yields (1 3)(1 2)(3 1) = (1 3 2), which has the same cycle structure as (1 2). Therefore, (1 2) and (1 3 2) are conjugate. Conjugating (1 2) by (2 3) yields (1 3), which is not equal to (1 2) or (1 3 2). Therefore, (1 2), (1 3 2), and (1 3) are all distinct conjugate classes.
Now, we consider the 3-cycles. Let's start with (1 2 3). Conjugating it by the identity permutation yields (1 2 3), which is not equal to (1 2 3). Conjugating it by (1 3) yields (1 3 2), which is the same as (1 2), which means that (1 2) and (1 3 2) are also conjugate to (1 2 3). Conjugating (1 2 3) by (2 3) yields (2 3 1), which is not equal to any of the other elements we've considered so far. Therefore, (1 2 3), (1 2), (1 3 2), and (2 3 1) are all distinct conjugate classes.
We repeat this process for the other elements of S3, finding that (1 3) and (2 1 3) are both conjugate to each other and form a distinct conjugate class.
In summary, the conjugate classes of S3 are:
- {e}, consisting of the identity permutation.
- {(1 2), (1 3 2), (1 3)}, consisting of the two transpositions and their conjugates.
- {(1 2 3), (1 2), (1 3 2), (2 3 1)}, consisting of the 3-cycles and their conjugates.
- {(1 3), (2 1 3)}, consisting of the two elements that are conjugate to each other.
To calculate the conjugate classes of the group S3, which is the symmetric group on three elements, we need to find the elements that are conjugate to each other.
First, let's write down all the elements of S3. The elements of S3 are the six permutations of the three elements: {1, 2, 3}. They are:
1. Identity element:
e = (1)(2)(3)
2. Transpositions:
(1,2), (2,3), (1,3)
3. 3-cycles:
(1,2,3), (1,3,2)
Next, we will determine the conjugate classes by finding the distinct orbits under the action of conjugation. Two elements are conjugate if there exists some element g ∈ S3 such that g⁻¹ag = b, where a and b are elements of S3.
Let's go through each element of S3 and find its conjugate(s):
1. Conjugate of the identity element: The identity element is always conjugate to itself. Therefore, the conjugacy class of the identity element is {e}.
2. Conjugates of transpositions:
- For (1,2):
The possible conjugates are:
- (1,2), (2,1)
Both (1,2) and (2,1) are in the same conjugacy class.
- For (2,3):
The possible conjugates are:
- (2,3), (3,2)
Both (2,3) and (3,2) are in the same conjugacy class.
- For (1,3):
The possible conjugates are:
- (1,3), (3,1)
Both (1,3) and (3,1) are in the same conjugacy class.
So, we have three distinct conjugacy classes for transpositions: {(1,2), (2,1)}, {(2,3), (3,2)}, and {(1,3), (3,1)}.
3. Conjugates of 3-cycles:
- For (1,2,3):
The possible conjugates are:
- (1,2,3), (2,1,3), (3,2,1)
All three are in the same conjugacy class.
- For (1,3,2):
The possible conjugates are:
- (1,3,2), (3,1,2), (2,3,1)
All three are in the same conjugacy class.
So, we have two distinct conjugacy classes for 3-cycles: {(1,2,3), (2,1,3), (3,2,1)} and {(1,3,2), (3,1,2), (2,3,1)}.
Therefore, the conjugate classes of the group S3 are: {e}, {(1,2), (2,1)}, {(2,3), (3,2)}, {(1,3), (3,1)}, {(1,2,3), (2,1,3), (3,2,1)}, {(1,3,2), (3,1,2), (2,3,1)}.