Question

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

6.$\mathit{k}\mathbf{=}\mathbf{⟨}\mathbf{3}\mathbf{⟩}\mathbf{;}\mathit{G}\mathbf{=}{\mathit{U}}_{\mathbf{32}}$

Short answer

The distinct right co-sets are$k1=k$ and $k5=\{5,7,13,15,21,23,29,31\}$.

Step-by-step solution

Determine given functions

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

Now let’s consider ${\mathrm{U}}_{32}$is the multiplicative subgroup of ${\mathrm{\mathbb{Z}}}_{32}$.

$\mathrm{k}=\{1,3,9,11,17,19,25,27\}$

Determine the list of co-sets

As order of ${\mathrm{U}}_{32}$is 16 and order of $k$is 2.

By applying the Lagrange’s theorem,

And from this role="math" localid="1654341353498" $k$has 2 co-sets, and these co-sets are,

role="math" localid="1654341357081" $k1=k$and role="math" localid="1654341381535" $k5=\{5,7,13,15,21,23,29,31\}$

Hence, the distinct co-sets are$k1=k$ and $k5=\{5,7,13,15,21,23,29,31\}$.