Question

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 are normal subgroups. This implies that$\mathrm{Q}$ and$\left\{1\right\}$ are normal subgroups. Since every elements of$\{1,-1\}$ commutes with every element of $\{1,-1\}$, this implies that$\{1,-1\}$ is normal subgroup.

Prove

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

By using the fact, it is observed that:

role="math" localid="1657296074197" $$\begin{gathered} ⟨\mathrm{i}⟩=\{1,-1,-\mathrm{i}\} \\ =\{1,-1\}⟨\mathrm{i}⟩ \end{gathered}$$

This implies thatrole="math" localid="1657296318409" $⟨\mathrm{i}⟩$is normal subgroup since we have already proved that $\{1,-1\}$is normal subgroup and is a subgroup of$\mathrm{Q}$.

Similarly,

role="math" localid="1657296178716" $$\begin{gathered} ⟨\mathrm{j}⟩=\{1,\mathrm{j},-1,-\mathrm{j}\} \\ =\{1,-1\}⟨\mathrm{j}⟩ \end{gathered}$$

This implies thatrole="math" localid="1657296343752" $⟨\mathrm{j}⟩$is normal subgroup since we have already proved that$\{1,-1\}$is normal subgroup and is a subgroup of$\mathrm{Q}$.

Similarly,

$$\begin{gathered} ⟨k⟩=\{1,k,-1,-k\} \\ =\{1,-1\}⟨k⟩ \end{gathered}$$

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

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