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Relationship between the quaternionic group and the quaternionic numbers?

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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?

asked Mar 7 in Chat by Dilaton (4,295 points) [ revision history ]
recategorized Mar 8 by Dilaton

I dont get it, is this some kind of trick question? The quaternion group is defined as taking the quaternion units as your group elements. Or in other words, if you allow your quaternion group elements to combine as basis elements in a vector space, you obtain quaternion numbers (you lose the original group structure though).

The quaternionic group is a finite subgroup of the multiplicative group of nonzero quaternionic numbers. That's how it has gotten its name. 

Zee (who mistakenly writes quarternionic group and numbers) doesn't claim that they are completely unrelated, just that they are two different kinds of algebraic structures.

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