Determine number of conjugacy class in $D_8$

Question

I’m using the formula that the number of conjugacy class is given to be $\frac{1}{|G|}\sum|C_{G}(g)|$, where $C_{G}(g)=\{h \in G ; gh=hg\}$, which is a special result by Burnside’s theorem.

I found that the number of conjugacy class in $D_8$ is 5, so to double check, I listed down all $C_{G}(g)=\{h \in G ; gh=hg\}$.

Let r be a rotation counter clockwise and m be a rotation in the x axis, 1 is the identity.

Well, $|C_G(1)|=|D_8|=16$, $|C_G(r)|= |C_G(r^2)|= |C_G(r^3)|= |C_G(r^5)|= |C_G(r^6)|= |C_G(r^7)|=|\{1,r,r^2,r^3,r^4,r^5,r^6,r^7\}|=8$,

$|C_G(r^4)|=|\{1,r,r^2,r^3,r^4,r^5,r^6,r^7,m\}|=9$ since $r^4m=mr^4$ Similarly, we can count for the reflections; $|C_G(m)|=|C_G(r^4m)|=|\{1,m\}|=2$, while the rest of the elements with any reflections only has one element, i.e $|C_G(r^nm)|=|\{1\}|=1,n \neq 0,4$

So the problem comes that my summation is 83, while my $|G|=16$. In this case I won’t get the number of conjugacy class to be 5. Did I do something wrong here? I just merely applied the definition...

Answer 1

$G = \{r,a|r^8 = a^2 = e, ara^{-1} = r^{-1}\}$

$e$ commutes with all elements and is in a conjugacy class all by itself.

$\{e\}$

Rotations fall into conjugacy classes that include their inverses.

$(r^n)(r^m)(r^{-n}) = r^m$

$(ar^n)(r^m)(ar^n)^{-1} = ar^nr^mr^{-n}a = r^{-m}$

$\{r, r^7\},\{r^2, r^6\},\{r^3, r^5\}, \{r^4\}$

Reflections:

$(r^n)(ar^m)(r^{-n}) = r^n (ar^{m-n}) = r^nr^{n-m}a = r^{2n-m}a = ar^{m-2n}$

and $(ar^n)(ar^m)(r^{-n}a) = ar^{2n-m}$

creating congugacy classes of $\{ar, ar^3, ar^5, ar^7\}$ and $\{a, ar^2, ar^4, ar^6\}$

That gives 7 conjugacy classes.

Counting the centralizers.

The identity commutes with everything.

Rotations commute with rotations.

$r^4$ commutes with every reflection, not just one reflection (as suggested above). Which means that $r^4$ commutes with everything.

Every reflection commutes with the identity, $r^4$, itself, and one other reflection.

$(ar^n)(r^{4-n}a) = (r^{4-n}a)(ar^n)=r^4$

$|C_G(e)| = 16\\ |C_G(r)| = 8 \text { times 6}\\ |C_G(r^4)| = 16\\ |C_G(a)| = 4 \text { times 8}$

$\frac {16\times 2 + 8\times 6 + 4\times 8}{16} = 7$

Answer 2

$D_n$ In general its class equation given by following

Case 1: n is odd

$Z(D_n)=${$e$}

$(n-1)/2$ classes of {$r^i,r^-i$} for $1\leq i\leq (n-1)/2$

$1$ class of {$sr^i|0\leq i \leq n-1$}

SO class equation $n=1+2.(n-1)/2+n$

Case 2:n is even

$Z(D_n)=${$e,r^{n/2}$}

$(n)/2-1$ classes of {$r^i,r^-i$} for $1\leq i\leq (n)/2-1$

$1$ class of {$sr^{2i}|0\leq i \leq n/2-1$}

$1$ class of {$sr^{2i+1}|0\leq i \leq n/2-1$}

SO class equation $n=1+1+2.(n/2-1)+n/2.1+n/2.1$

In specific: $D_8$

$1+1+2+2+2+4+4=16 $is class equation