Question

(a) List all the cyclic subgroups of ${\mathbf{D}}_4$.

Short answer

The cyclic subgroups of ${\mathrm{D}}_4$ are$⟨{\mathrm{r}}_0⟩,⟨{\mathrm{r}}_1⟩,⟨{\mathrm{r}}_2⟩,⟨\mathrm{d}⟩,⟨\mathrm{h}⟩,⟨\mathrm{t}⟩,⟨\mathrm{v}⟩$ .

Step-by-step solution

Cyclic Subgroup

Every element x in a group G, form a subgroup of all integer powers$⟨\mathrm{x}⟩=\{{\mathrm{x}}^{\mathrm{k}},\mathrm{k}\in \mathrm{\mathbb{Z}}\}$ is called a cyclic subgroup of x.

Elements in D4

The set of elements in${\mathrm{D}}_4$ is ${\mathrm{D}}_4=\{{\mathrm{r}}_0,{\mathrm{r}}_1,{\mathrm{r}}_2,{\mathrm{r}}_3,\mathrm{d},\mathrm{h},\mathrm{t},\mathrm{v}\}$.

Cyclic Subgroups of D4

For element ${\mathrm{r}}_0\in {\mathrm{D}}_4$,$⟨{\mathrm{r}}_0⟩=\left\{1\right\}$, for element ${\mathrm{r}}_1\in {\mathrm{D}}_4$, $⟨{\mathrm{r}}_1⟩=\{{\mathrm{r}}_0,{\mathrm{r}}_1,{\mathrm{r}}_2,{\mathrm{r}}_4\}$, for element ${\mathrm{r}}_2\in {\mathrm{D}}_4$, $⟨{\mathrm{r}}_2⟩=\{{\mathrm{r}}_2,{\mathrm{r}}_4\}$, for element ${\mathrm{r}}_3\in {\mathrm{D}}_4$,$⟨r_3⟩=\{r_0,r_1,r_2,r_4\}$.

Then for element $\mathrm{d}\in {\mathrm{D}}_4$, $⟨\mathrm{d}⟩=\{{\mathrm{r}}_0,\mathrm{d}\}$, for element $\mathrm{h}\in {\mathrm{D}}_4$, $⟨\mathrm{h}⟩=\{{\mathrm{r}}_0,\mathrm{h}\}$, for element $\mathrm{t}\in {\mathrm{D}}_4$,$⟨\mathrm{t}⟩=\{{\mathrm{r}}_0,\mathrm{t}\}$, for element $\mathrm{v}\in {\mathrm{D}}_4$,$⟨\mathrm{v}⟩=\{{\mathrm{r}}_0,\mathrm{v}\}$.

Therefore, there are 6 different cyclic subgroups which are $⟨{\mathrm{r}}_0⟩,⟨{\mathrm{r}}_1⟩,⟨{\mathrm{r}}_2⟩,⟨\mathrm{d}⟩,⟨\mathrm{h}⟩,⟨\mathrm{t}⟩,⟨\mathrm{v}⟩$.