# Class Equation for Conjugation Actions in A_4, D_8, and D_{10}

• mathusers
In summary, the conversation discussed the Class Equation for group actions by conjugation, specifically for the groups A_4, D_8, and D_{10}. The conversation then focused on the case of D_8 and provided a detailed calculation of the orbits for the group. The conversation concluded by asking for a description of the set-theorem and numerical forms of the Class Equation for D_8.
mathusers
hi, any hints to give me a good idea on how to solve this question would be greatly appreciated:

Question:
let G be a group acting on itself by conjugation, $g . x = gxg^{-1}$

Describe the set-theorem and numerical forms of the Class Equation for these actions explicitly when

(i) $G = A_4$
(ii) $G = D_8$
(iii) $G = D_{10}$

thanx :)

ok let me try part (ii) $G = D_8$

my working:
------------------------------------------------------------------------
Firstly $D_8 = [1,x,x^2,x^3,y,xy,x^2y,x^3y]$
$x^4 = 1, y^2 = 1, yx = x^3y$

If we calculate the orbits then we have:
orbit of <1> = {1}

<x>:
$(1)x(1^{-1}) = x$,
$(x)x(x)^{-1} = x$,
$(x^2)x(x^2)^{-1} = x$,
$(x^3)x(x^3)^{-1} = x$,
$(y)x(y)^{-1} = x^2yy=x^2$,
$(xy)x(xy)^{-1} = xyxx^3y=xx^2yx^3y=x^6=x^2$,
$(x^2y)x(x^2y)^{-1} = x^2yxx^2y=x^2x^2yx^2y=x^4yx^2y=x^6yy=x^6y^2=x^6=x^2$,
$(x^3y)x(x^3y)^{-1} = x^3yxxy=x^3x^2yxy=x^3x^2x^2yy=x^7y^2=x^7=x^3$,
so orbit of <x> = <x^2> = <x^3> = {$x,x^2,x^3$}

similarly,
<y>:
$(1)y(1^{-1}) = y$,
$(x)y(x)^{-1} = x^3y$,
$(x^2)y(x^2)^{-1} = x^2y$,
$(x^3)y(x^3)^{-1} = xy$,
$(y)y(y)^{-1} = y$,
$(xy)y(xy)^{-1} = x^3y$,
$(x^2y)y(x^2y)^{-1} = x^2y$,
$(x^3y)y(x^3y)^{-1} =xy$,
so orbit of <y> = <xy> <x^2y> = <x^3y> = {$y,xy,x^2y,x^3y$}

concluding, the if $D_8$ cuts on itself by conjugation, the orbits are:
<1> = {1}
<x> = <$x^2$> = <$x^3$> = {$x,x^2,x^3$}
<y> = <$xy$> = <$x^2y$> = <$x^3y$> = {$y,xy,x^2y,x^3y$}
------------------------------------------------------------------------

i believe I've done the hard part: so from here how would i descibe explicitly, the set theorem and numerical forms of the class equation for D_8 ? thanks :)

## 1. What is the definition of the class equation for conjugation actions in A4, D8, and D10?

The class equation for conjugation actions in a group is an equation that shows the number of elements in the group divided into classes, where each class consists of elements that are conjugate to each other.

## 2. How is the class equation calculated for these specific groups?

The class equation for A4, D8, and D10 can be calculated by first finding the number of elements in the group, then dividing that number by the number of elements in each conjugacy class.

## 3. What are the conjugacy classes in A4, D8, and D10?

The conjugacy classes in these groups are the sets of elements that are equivalent under the conjugation action. In A4, there are three conjugacy classes: the identity element, the 3-cycles, and the products of two disjoint 2-cycles. In D8, there are four conjugacy classes: the identity element, the reflections, the rotations, and the reflections combined with rotations. In D10, there are five conjugacy classes: the identity element, the reflections, the rotations, the reflections combined with rotations, and the rotations combined with reflections.

## 4. How do these class equations differ from each other?

The class equations for these groups differ in the number of elements and the number of conjugacy classes. A4 has 12 elements and 3 conjugacy classes, D8 has 8 elements and 4 conjugacy classes, and D10 has 10 elements and 5 conjugacy classes.

## 5. What is the significance of the class equation for conjugation actions in these groups?

The class equation is significant because it helps us understand the structure of the group and the relationships between its elements. It also allows us to calculate the size of each conjugacy class, which can be useful in solving problems and proving theorems in group theory.

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