Quaternion Group as Permutation Group I was recently, for the sake of it, trying to represent Q8, the group of quaternions, as a permutation group. I couldn't figure out how to do it.
So I googled to see if somebody else had put the permutation group on the web, and I came across this:
http://mathworld.wolfram.com/PermutationGroup.html

Not all groups are representable as permutation groups. For example, the quaternion group cannot be represented in terms of permutations. 

This strikes me as a very odd statement, because I quickly checked this:
http://mathworld.wolfram.com/CayleysGroupTheorem.html

Every finite group of order n can be represented as a permutation group on n letters

So it seems Q8 should be representable a sa permutation group on 8 letters.
How can these two quotes be reconciled?

Follow Cayley's embedding: write down the elements of $Q_8=\{1,-1,i,-i,j,-j,k,-k\}$ as an ordered set, and left-multiply each element with successively with each element of this set - this yields a permutation, e.g. multiplication from the left with $i$, gives you that the ordered set $(1,-1,i,-i,j,-j,k,-k)$ goes to $(i,-i,-1,1,k,-k,-j,j)$, which corresponds to the permutation $(1324)(5768)$. Etc. Can you take it from here? So it can be done and the statement on the WolframMathWorld - Permutation Groups page must be wrong.
