Question

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

$\mathit{k}\mathbf{=}\mathbf{\{}{\mathbf{r}}_{\mathbf{0}}\mathbf{,}\mathbf{\upsilon }\mathbf{\}}\mathbf{:}\mathit{G}\mathbf{=}{\mathit{D}}_{\mathbf{4}}$[The operation table for${\mathit{D}}_{\mathbf{4}}$ is in Example 5 of Section 7.1

or 7.1.A.]

Short answer

The distinct co-sets are,$k{r}_0=k,k{r}_1=\{r_1,t\},k{r}_2=\{r_2,h\},k{r}_3=\{r_3,d\}$

Step-by-step solution

Determine given functions

Consider the given function$G$ is the group and$k$ is the subgroup.

Now, let $D_4$is the dihedral group.

Determine the list of co-sets

As order of$D_4$is 8 and order of$k$is 2.

By applying the Lagrange’s theorem,

And from this $k$has 4 co-sets, and these co-sets are,

$k{r}_0=k,k{r}_1=\{r_1,t\},k{r}_2=\{r_2,h\},k{r}_3=\{r_3,d\}$

Hence, the required co-sets are $k{r}_0=k,k{r}_1=\{r_1,t\},k{r}_2=\{r_2,h\},k{r}_3=\{r_3,d\}$.