The quaternion group $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$ inside the nonzero quaternions

Let $\mathbb{H}$ be the quaternions, with the basis quaternions $1=(1,0,0,0)$, $i=(0,1,0,0)$, $j=(0,0,1,0)$, $k=(0,0,0,1)$ and the real embedding $\lambda \mapsto \widehat{\lambda }$ of The quaternions $\mathbb{H}$: real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1,i,j,k$. Write

$n:=\widehat{-1}=(-1,0,0,0),-x:=nx(x\in \mathbb{H}),$

so that $-x=(-x_0,-x_1,-x_2,-x_3)$ by the formula $\widehat{\lambda }x=x\widehat{\lambda }=(\lambda x_0,\lambda x_1,\lambda x_2,\lambda x_3)$ recorded in The quaternions $\mathbb{H}$: real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1,i,j,k$.

By $\mathbb{H}$ is a division ring that is not commutative, hence not a field: $q^{-1}=\bar{q}\mathrm{/}N(q)$ for $q\mathrm{\ne }0$, while $ij=k$ and $ji=-k$ the set $\mathbb{H}\setminus \{0_{\mathbb{H}}\}$ is a group under quaternion multiplication (Group and abelian group); write it ${\mathbb{H}}^{\times }$. The quaternion group is the subset

$Q_8:=\{1,-1,i,-i,j,-j,k,-k\}\subseteq {\mathbb{H}}^{\times }\mathrm{.}$

That $Q_8$ is a subgroup of ${\mathbb{H}}^{\times }$ (Subgroup), that it has exactly eight elements, and that $-1$ is its only element of order $2$ are proved in $Q_8$ is a subgroup of ${\mathbb{H}}^{\times }$ with eight elements, and $-1$ is its only element of order $2$ and are not assumed here.

• Nothing is adjoined to $\mathbb{H}$. The eight listed quaternions are particular quadruples of real numbers and the operation is the multiplication already defined on $\mathbb{H}$; no new multiplication table is postulated, and every product below is read off the table $i^2=j^2=k^2=-1$, $ij=k$, $jk=i$, $ki=j$, $ji=-k$, $kj=-i$, $ik=-j$ that The quaternions $\mathbb{H}$: real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1,i,j,k$ derives from its product formula.

• $-1$ is not the additive inverse taken on trust. The abbreviation $-x$ is defined here as the product $nx$ with the specific quaternion $n=(-1,0,0,0)$; that this coincides with componentwise negation is the displayed consequence of the product formula, not a separate convention.