Question

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

9.$\mathit{H}\mathbf{=}\mathbf{⟨}\mathbf{3}\mathbf{⟩}\mathbf{;}\mathit{G}\mathbf{=}{\mathit{\mathbb{Z}}}_{\mathbf{20}}$

Short answer

$k$has 1 distinct 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 ${\mathbb{Z}}_{20}$is the group of integers modulo 20.

$k=\{0,1,3,4,6,7,9,10,12,13,14,15,16,17,18,19\}$

Determine index

As order of${\mathbb{Z}}_{20}$ is 20 and order of$k$ is 20.

By applying the Lagrange’s theorem,

And from this$k$ has 1 distinct co-sets.