Relationship between the quaternionic group and quaternionic numbers? 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 dividion 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?

It's silly to say that they are completely unrelated. If you take just those quaternions for which precisely one of $a,b,c,d$ is non-zero, and the one that is non-zero is either 1 or $-1$, you obtain the quaternionic group. On the other hand, if you use the elements of the quaternionic group as the basis of an 8-dimensional vector space over $\mathbb R$, and then add relations to make $-x$ the negation of $x$ for each element of the group (creating a 4-dimensional vector space), and define a multiplication on that vector space by using the multiplication rules of the quaternionic group, you get precisely the algebra of the quaternions.
Presumably, your book just wanted to emphasize that you should not confuse the two structures, as one is a group under multiplication, while one is an ($\mathbb R$-, or $\mathbb C$-)algebra.
