Quaternion group

In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication. It is given by the group presentation In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication. It is given by the group presentation where e is the identity element and e commutes with the other elements of the group. The quaternion group Q8 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 can be realized as a subset of the split-quaternions in the same way that Q8 can be viewed as a subset of the quaternions. The Cayley table (multiplication table) for Q8 is given by: The quaternion group has the unusual property of being Hamiltonian: every subgroup of Q8 is a normal subgroup, but the group is non-abelian. Every Hamiltonian group contains a copy of Q8. The quaternion group Q8 is one of the two smallest examples of a nilpotent non-abelian group, the other being the dihedral group D4 of order 8. The quaternion group Q8 has five irreducible representations, and their dimensions are 1,1,1,1,2. The proof for this property is not difficult, since the number of irreducible characters of Q8 is equal to the number of its conjugacy classes, which is five ( { e }, { e }, { i, i }, { j, j }, { k, k } ).