Question

In Exercises 7-11 is $\mathit{G}$ a group and$\mathit{H}$is a subgroup of G. Find the index $\mathbf{[}\mathbf{G}\mathbf{:}\mathbf{H}\mathbf{]}$.

$\mathit{H}\mathbf{=}\mathbf{\{}{\mathbf{r}}_{\mathbf{0}}\mathbf{,}{\mathbf{r}}_{\mathbf{2}}\mathbf{\}}\mathbf{;}\mathit{G}\mathbf{=}{\mathit{D}}_{\mathbf{4}}$

Short answer

$k$has 4 co-sets.

Step-by-step solution

Determine given functions

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

Now let’s consider$D_4$ is the dihedral group and$k$ is the subgroup $\mathbf{\{}{\mathbf{r}}_{\mathbf{0}}\mathbf{,}\mathbf{v}\mathbf{\}}$.

Determine index

As order of $D_4$is 8 and order of role="math" localid="1654341840509" $k$is 2.

By applying Lagrange’s theorem,

And from this$k$ has 4 distinct co-sets.