Question

List all the cyclic subgroups of the quaternion group (Exercise 16 of Section 7.1 ).

Short answer

The cyclic subgroups are $\mathrm{Q}\left\{1\right\},\{1-1\}⟨\mathrm{i}⟩=\{1,-1-\mathrm{i}\},⟨\mathrm{j}⟩=\{1,\mathrm{j},-1,-\mathrm{j}\}⟨\mathrm{k}⟩=\{1,\mathrm{k},-1,\mathrm{k}\}$.

Step-by-step solution

Write the elements of Quaternion group

The elements in quaternion group are$\{1,\mathrm{i},\mathrm{j},\mathrm{k},-1,-\mathrm{i},-\mathrm{j},-\mathrm{k}\}.$

Write cyclic subgroups

The cyclic subgroups of the quaternion group are:

$$\begin{gathered} \mathrm{Q},\left\{1\right\},\{1,-1\} \\ ⟨\mathrm{i}⟩=\{1,\mathrm{i},-1,-\mathrm{i}\} \\ ⟨\mathrm{j}⟩=\{1,\mathrm{j},-1,-\mathrm{j}\} \\ ⟨\mathrm{k}⟩=\{1,\mathrm{k},-1,-\mathrm{k}\} \end{gathered}$$

Thus, the cyclic subgroup are

$\mathrm{Q}\left\{1\right\},\{1-1\}⟨\mathrm{i}⟩=\{1,-1-\mathrm{i}\},⟨\mathrm{j}⟩=\{1,\mathrm{j},-1,-\mathrm{j}\}⟨\mathrm{k}⟩=\{1,\mathrm{k},-1,\mathrm{k}\}.$