Question

List all subgroups of${\mathit{\mathbb{Z}}}_{\mathbf{20}}\mathbf{/}\mathit{K}$, where$\mathit{K}\mathbf{=}\mathbf{\{}\mathbf{0}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{8}\mathbf{,}\mathbf{12}\mathbf{,}\mathbf{16}\mathbf{\}}$.

Short answer

The subgroups of ${\mathbb{Z}}_{20}/K$are$\left\{0\right\},{\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$.

Step-by-step solution

Referring to Lagrange’s Theorem

If K is a subgroup ofa finite group G. Then the order of K divides the order of G. In particular, $\mathbf{\left|}\mathbf{G}\mathbf{\right|}\mathbf{=}\mathbf{\left|}\mathbf{K}\mathbf{\right|}\mathbf{[}\mathbf{G}\mathbf{:}\mathbf{K}\mathbf{]}$.

Finding all subgroups of ℤ12/H

The order of${\mathbb{Z}}_{20}/K$ can be given as:

role="math" localid="1659509015769" $\begin{array}{c}O({\mathbb{Z}}_{20}/K)=\left|{\mathbb{Z}}_{20}\right|/\left|K\right|\\ =\frac{20}{5}\\ =4\end{array}$

Since the order of${\mathbb{Z}}_{20}/K$ is 4, according toLagrange’s Theorem all the subgroups${\mathbb{Z}}_{20}/K$ of must have order as multiples of 4.

Hence, the subgroups are$\left\{0\right\},{\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$.