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

It is proved that all the subgroups are normal.

Step-by-step solution

Quaternion group

From exercise 4, we know that if$\mathrm{G}$ is the group then, and$\mathrm{G}$ are normal subgroups. This implies that $\mathrm{Q}$and$\left\{1\right\}$ are normal subgroups. Since every elements of$\left\{1,-1\right\}$ commutes with every element of $\left\{1,-1\right\}$, this implies that$\left\{1,-1\right\}$ is normal subgroup.

Prove

Use the fact that states that if $\mathrm{K}$be a normal subgroup of a group $\mathrm{G}$. Then, $\mathrm{HK}=\left\{\mathrm{hk}:\mathrm{h}\in \mathrm{H},\mathrm{k}\in \mathrm{K}\right\}$is a subgroup, where $\mathrm{H}$is a subgroup of$\mathrm{G}$.

By using the fact, it is observed that:

role="math" localid="1655884953178"

This implies that is normal subgroup since we have already proved that $\left\{1,-1\right\}$is normal subgroup and is a subgroup of $\mathrm{Q}$.

Similarly,

This implies that is normal subgroup since we have already proved that$\left\{1,-1\right\}$ is normal subgroup and is a subgroup of $\mathrm{Q}$.

Similarly,

This implies that is normal subgroup since we have already proved that $\left\{1,-1\right\}$is normal subgroup and is a subgroup of $\mathrm{Q}$.

Hence, it is proved that all the subgroups are normal.