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 $\mathit{G}$.

5.$\mathit{k}\mathbf{=}\mathbf{\left\{}\begin{array}{cc}\mathbf{1}\mathbf{,}& \mathbf{17}\end{array}\mathbf{\right\}}\mathbf{;}\mathit{G}\mathbf{=}{\mathit{U}}_{\mathbf{32}}$

Short answer

The distinct right co-sets are:

$\begin{array}{l}k1=k,k3=\{2,19\},k5=\{5,21\},k7=\{7,23\},k9=\{9,25\}\\ k11=\{11,27\},k13=\{13,29\},k15=\{15,31\}\end{array}$

Step-by-step solution

Determine given functions

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

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

Determine the list of co-sets 

As order of $U_{32}$is 16 and order of role="math" localid="1654339503282" $k$is 8.

By applying Lagrange’s theorem,

And from this $k$has 8 co-sets, and these co-sets are,

$\begin{array}{l}k1=k,k3=\{2,19\},k5=\{5,21\},k7=\{7,23\},k9=\{9,25\}\\ k11=\{11,27\},k13=\{13,29\},k15=\{15,31\}\end{array}$

Hence, the distinct right co-sets are

$k1=k,k3=\{2,19\},k5=\{5,21\},k7=\{7,23\},k9=\{9,25\},k11=\{11,27\},k13=\{13,29\}$and$k15=\{15,31\}$.