Question

$\mathbf{G}$is a group and $\mathit{k}$is a subgroup of $\mathbf{G}$. List the distinct right co-sets of $\mathit{k}$in $\mathit{G}$.

3.$\mathit{k}\mathbf{=}\mathbf{\{}{\mathbf{r}}_{\mathbf{0}}\mathbf{,}{\mathbf{r}}_{\mathbf{1}}\mathbf{,}{\mathbf{r}}_{\mathbf{2}}\mathbf{,}{\mathbf{r}}_{\mathbf{3}}\mathbf{\}}\mathbf{:}\mathit{G}\mathbf{=}{\mathit{D}}_{\mathbf{4}}$

Short answer

The distinct right co-sets are,$k{r}_0=k$ and $kd=\{d,h,t,v\}$.

Step-by-step solution

Determine given functions

Considering the given functions, $G$is the group and $k$ is the subgroup.

Now let’s consider $D_4$is the dihedral group.

Determine list of co-sets

As order of$D_4$ is 8 and order ofrole="math" localid="1654337956721" $k$ is 4.

By applying the Lagrange’s theorem,

And from this$k$ has 2 co-sets, and these co-sets are,

$k{r}_0=k$and $kd=\{d,h,t,v\}$.

Hence, the right co-sets are$k{r}_0=k$ and $kd=\{d,h,t,v\}$.