[Math] Number of conjugacy classes of a Dihedral group

dihedral-groupsgroup-theory

How do you find the number of conjugacy classes of a Dihedral group?
Say for D11 for example. I know by Lagrange each conjugacy class has order 1, 2, or 11.
For smaller n, it can sometimes just be broken up since the sum of the orders of the conjugacy classes equals the order of the group, 22.
I'm not sure where to go from here…

Best Answer

Calculating in a Dihedral group is not very difficult.
$D_n = (a, b|a^2=b^n = (ab)^2 = e)$ and thus
$D_n = \{e, b, b^2, ...,b^{n-1}, a, ab, ab^2, ..., ab^{n-1}\}$
where multiplication can be derived by observing that $ba=ab^{n-1}$.
You can now show that the elements of $D_n$ have orders equal to the orders of the cyclic subgroup of order $n$ plus $n$ elements of order $2$.

To compute conjugacy classes observe

  • $b(ab^j)b^{n-1}=ab^{j-2}$ and
  • $a(a^{\epsilon}b^j)a = a^{\epsilon}b^{n-j} $ where $\epsilon = 0$ or $1$.
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