Quaternion group

Definition by presentation
The quaternion group has the following presentation:

$$\langle i,j,k \mid i^2 = j^2 = k^2 = ijk \rangle$$

The identity is denoted $$1$$, the common element $$i^2 = j^2 = k^2 = ijk$$ is denoted $$-1$$, and the elements $$i^3, j^3, k^3$$ are denoted $$-i,-j,-k$$ respectively.

Verbal definitions
The quaternion group is a group with eight elements, which can be described in any of the following ways:


 * It is the group comprising eight elements $$1,-1,i,-i,j,-j,k,-k$$ where 1 is the identity element, $$(-1)^2 = 1$$ and all the other elements are squareroots of $$-1$$, such that $$(-1)i = -i, (-1)j = -j, (-1)k= -k$$ and further, $$ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j$$ (the remaining relations can be deduced from these).
 * It is the with parameter 2, viz $$Dic_2$$.
 * It is the member of family::Fibonacci group $$F(2,3)$$.

Smallest of its kind

 * This is a non-abelian nilpotent group of smallest possible order, along with dihedral group:D8.
 * This is a non-abelian Dedekind group (or Hamiltonian group) of smallest possible order. Dedekind means that every subgroup is normal.

Different from others of the same order

 * It is the only non-abelian Dedekind group of its order.
 * It is the only non-abelian T-group of its order.
 * It is the only group of its order for which the rank (in the sense of the maximum possible rank of an abelian subgroup) is strictly smaller than the minimum size of generating set: For this group, the former is 1 and the latter is 2.