Difficult# Conjugacy Classes: Dihedral Group with $2n$ Elements

ABSALG-CH2601

Let $D_n$, $n\ge 2$, be the dihedral group with $2n$ elements and neutral element $e$. It has the presentation in terms of generators and relators:

$$\langle a,b \mid a^n=b^2=e; \; aba=b^{-1}\rangle$$

1) When $n$ is odd, the conjugacy class of $a^k$ for $k=1,\ldots,n-1$ is

.

2) When $n$ is even, the conjugacy class of $a^k$ for $k=1,\ldots, n-1$ is

when $k\not=\cfrac{n}2$. The conjugacy class of $a^{n/2}$ is

.

3) If $n$ is odd, the conjugacy class of $b$ consists of

.

4) If $n$ is even, the conjugacy class of $b$ consists of

.

5) When $n$ is odd, the group $D_n$ is the disjoint union of

distinct conjugacy classes.

6) When $n$ is even, the group $D_n$ is the disjoint union of

distinct conjugacy classes.