Quaternions

General

The quaternion group can be represented by eight matrices over the complex numbers. It is a non abelian group. The matrices consist of the 4 diagonal matrices (i,-i), (-1,-1), (-i,i),(1,1) and 4 antidiagonal matrices (0,1,-1,0), (0,-1,1,0), (0,i,i,0), (0,-i,-i,0).

Order

8

Elements

1 of order 2, 6 of order 4

Generators

Let a be the diagonal matrix (i,-i) and b the antidiagonal matrix (0,1,-1,0) then the quaternion group is generated by these two elements. The elements may be represented as 1, a, , , b, ab, and .

Subgroups

6 subgroups, all of which are normal and cyclic.

Factor groups

, 1, ,