Quaternion group
In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication. It is often denoted by Q or Q8, and is given by the group presentation
where 1 is the identity element and −1 commutes with the other elements of the group.
Compared to dihedral group
The Q8 group has the same order as the dihedral group D4, but a different structure, as shown by their Cayley and cycle graphs:
The dihedral group D4 arises in the split-quaternions in the same way that Q8 lies in the quaternions.
Cayley table
The Cayley table (multiplication table) for Q is given by:
The multiplication of the six imaginary units {±i, ±j, ±k} works like the cross product of unit vectors in three-dimensional Euclidean space.
Properties
The quaternion group has the unusual property of being Hamiltonian: every subgroup of Q is a normal subgroup, but the group is non-abelian. Every Hamiltonian group contains a copy of Q.
