Conjugate classes of A3
The conjugate classes of A3, the alternating group of degree 3, are the following:
1. {(), (1 2)(3), (1 3)(2)} - This is the identity class, which contains the identity element () and two elements that are conjugate to each other, namely (1 2)(3) and (1 3)(2). These two permutations switch the positions of two elements while leaving the third in place.
2. {(1 2 3), (1 3 2)} - This class contains two elements that are conjugate to each other, namely (1 2 3) and (1 3 2). These two permutations cyclically permute the three elements.
Note that since A3 has only three elements, there cannot be any other conjugacy classes.
What's the value of
1) I(123)(I^-1)
2) (132)(123)(231)
1) I(123)I^-1 = (123), since I (the identity permutation) and I^-1 (also the identity permutation) do not affect the conjugation.
2) We can simplify the product (132)(123)(231) as follows:
(132)(123)(231) = (132)(1)(23)(231) (inserting two factors of the identity permutation)
= (132)(23)(231) (rearranging factors)
= (12)(3) (multiplying out)
Therefore, (132)(123)(231) = (12)(3) in cycle notation.
To determine the conjugacy classes of the alternating group A3, we can use the following steps:
Step 1: Determine the elements of A3.
The alternating group A3 consists of the even permutations of the three elements {1, 2, 3}. These permutations are:
- The identity permutation: e = (1)(2)(3)
- Three 2-cycles: (12), (23), (13)
- One 3-cycle: (123)
Step 2: Find the centralizer of each element.
The centralizer of an element g in A3 is defined as the set of elements that commute with g. In other words, it is the set of elements h in A3 such that gh = hg.
- For the identity permutation e, the centralizer is the entire group A3 itself. The centralizer of e is {e, (12), (23), (13)}.
- For the 2-cycles (12), (23), and (13), the centralizer of each is just the respective 2-cycle itself. The centralizers are:
- Centralizer of (12): {(12)}
- Centralizer of (23): {(23)}
- Centralizer of (13): {(13)}
- For the 3-cycle (123), the centralizer is the entire group A3 itself. The centralizer of (123) is {e, (12), (23), (13), (123), (132)}.
Step 3: Group the elements based on their conjugacy classes.
Elements in A3 that have the same centralizer are in the same conjugacy class.
- The identity permutation e and the 3-cycle (123) have the same centralizer and are in the same class. Hence, {(1)(2)(3), (123)} forms a conjugacy class.
- The 2-cycles (12), (23), and (13) each have a centralizer that is just the respective 2-cycle itself, so they form their individual conjugacy classes. Hence, {(12)}, {(23)}, and {(13)} are three separate conjugacy classes.
Therefore, the conjugacy classes of A3 are:
{(1)(2)(3), (123)}, {(12)}, {(23)}, and {(13)}.