Conjugate classes of A3

Answer 1

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.

Answer 2

What's the value of
1) I(123)(I^-1)
2) (132)(123)(231)

Answer 3

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.