Question

List all the subgroups of ${\square }_{12}$. Do the same for${\square }_{20}$

Short answer

The subgroups of Z₁₂and Z₂₀ are listed

Step-by-step solution

Required Theorem

Theorem 7.17:

Every subgroup of a cyclic group is itself cyclic.

List all the subgroups of □12

The subgroups are listed below:

$$\begin{gathered} {\mathrm{H}}_1=\{0,2,4,6,8\} \\ {\mathrm{H}}_2=\{0,3,6,9\} \\ {\mathrm{H}}_3=\{0,4,8\} \\ {\mathrm{H}}_4=\{0,6\} \\ {\mathrm{H}}_5=\left\{0\right\} \end{gathered}$$

List all the subgroups of □20

The subgroups are listed below:

$$\begin{gathered} {\mathrm{H}}_1=\{0,2,4,6,8,10,12,14,16,18\} \\ {\mathrm{H}}_2=\{0,4,8,12,16\} \\ {\mathrm{H}}_3=\{0,5,10,15\} \\ {\mathrm{H}}_4=\{0,10\} \\ {\mathrm{H}}_5=\left\{0\right\} \end{gathered}$$