Computing the conjugacy classes of D20(dihedral group of order 20)

[playa007]

Homework Statement

To compute the conjugacy classes of D20

Homework Equations

The class equation, center of the group, group order, Lagrange's theorem that number of elements in conjugacy classes divides order of group

The Attempt at a Solution

i'm struggling to understand how to compute the conjugacy class of this group; i understand the definition of a conjugacy class but i just need some help on how to compute its conjugacy classes

Details on the steps required to do this is very much appreciated

 

[Nino MMP]

. The group D20 is a dihedral group, and has order 20. This means that it has 20 elements, and so there can be at most 20 conjugacy classes.To compute the conjugacy classes of D20, we will use the class equation. The class equation states that the sum of the sizes of the conjugacy classes must equal the order of the group.The first step is to determine the size of the center of the group. The center of D20 is Z(D20) = {e}, which has size 1.Next, we need to determine the number of conjugacy classes in D20. Note that by Lagrange's theorem, the number of elements in each conjugacy class must divide the order of the group, which is 20. So, the number of conjugacy classes in D20 must be 1, 2, 4, 5, 10, or 20.We can now use the class equation to calculate the number of elements in each conjugacy class. Since the center has size 1, the remaining 19 elements must be distributed among the other conjugacy classes. If there are 1 conjugacy classes, then all 19 elements must belong to one class, and so that class must have size 19.If there are 2 conjugacy classes, then one class must have size 10 and the other must have size 9.If there are 4 conjugacy classes, then one class must have size 5 and the remaining 3 classes must each have size 4.If there are 5 conjugacy classes, then one class must have size 4 and the remaining 4 classes must each have size 3.If there are 10 conjugacy classes, then each class must have size 2.Finally, if there are 20 conjugacy classes, then each class must have size 1.To determine which of these possibilities is correct, we will use a computer algebra package like GAP (Groups, Algorithms, and Programming). In GAP, we can enter the following command:SizeClasses( DihedralGroup(20) );This will tell us that the conjugacy classes in D20 are of size 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2,