Question

List the distinct conjugacy classes of the group ${\mathit{D}}_{\mathbf{4}}$.

Short answer

Conjugacy classes of$D_4$ is given by

$\left\{\left\{1\right\},\left\{r^2\right\},\left\{s,r^2,s\right\},\left\{r,r^3\right\},\left\{rs,r^{3}s\right\}\right\}$

Step-by-step solution

Structure of D4

Let Order of ${\mathit{D}}_{\mathbf{4}}\mathbf{=}\mathbf{8}$

${\mathit{D}}_{\mathbf{4}}\mathbf{=}\left\{1,r,r^2,r^3,s,sr,s{r}^2,s{r}^3\right\}$

Conjugacy class

Let G be a group and $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{G}$. We say that a is conjugate to b if there exist $x\in G$such that $\mathit{b}\mathbf{=}{x}^{-1}\mathit{a}\mathit{x}$ .

The conjugacy class of an element a consists of all the elements in G that are conjugate to a

Also we know that conjugacy is equivalence class

There are five conjugacy classes of$D_4$and is given by,

$\left\{\left\{1\right\},\left\{r^2\right\},\left\{s,r^2,s\right\},\left\{r,r^3\right\},\left\{rs,r^{3}s\right\}\right\}$