The quaternionic group $\mathcal{Q}$ consists of the elements $1$, $-1$, $i$, $-i$,$j$,$-j$,$k$,$-k$ that satisfy the multiplication rules

$$i^2=j^2=k^2=-1$$

$$ ij=-ji=k$$

$$jk=-kj=i$$

$$ki=-ik=j$$

The quaternionic numbers $$a+ib+cj+dk$$ form a division algebra.

In Group Theory in a Nutshell on p61 A.Zee writes that those two structures are completely unrelated, but I almost cant swallow this.

Are the quaternionic group and the quaternionic numbers really completely unrelated?