Question

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

(b) Show that each of the subgroups in part (a) is normal.

Short answer

The cyclic subgroups are.

Step-by-step solution

Write the elements of Quaternion group

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

Write cyclic subgroups

The cyclic subgroups of the quaternion group are:

Thus, the cyclic subgroup are .