Question

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

10.$\mathit{H}$ is the subgroup generated by 12 and 20;$\mathit{G}\mathbf{=}{\mathit{\mathbb{Z}}}_{\mathbf{40}}$ .

Short answer

$k$has 4 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 ${\mathbb{Z}}_{40}$is the group of integers modulo 20, $k$is subgroup $⟨12,20⟩$generated by 12 and 20.

${\mathbb{Z}}_{40}$is cyclic group and $k$is subgroup, thus $k$is also cyclic.

The cyclic subgroup$k$ is the greatest common divisor of 12 and 20 is 4. Hence,$k=⟨4⟩=\{0,4,8,12,16,20,24,28,32,36\}$

Determine index

As order of${\mathbb{Z}}_{20}$ is 40 and order of$k$ is 10.

By applying the Lagrange’s theorem,

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