Question

2(b) List at least one subgroup of ${\mathbf{D}}_4$ that is not cyclic.

Short answer

The groups${\mathrm{D}}_4$ itself is a subgroup which is not cyclic.

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

The cyclic subgroups of ${\mathrm{D}}_4$are $⟨{\mathrm{r}}_0⟩,⟨{\mathrm{r}}_1⟩,⟨{\mathrm{r}}_2⟩,⟨\mathrm{d}⟩,⟨\mathrm{h}⟩,⟨\mathrm{t}⟩,⟨\mathrm{v}⟩$where the ${\mathrm{D}}_4$group itself is a subgroup which is not cyclic.

Therefore, the groups${\mathrm{D}}_4$ itself is a subgroup which is not cyclic.