Generalized 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 e is the identity element and e commutes with the other elements of the group.
The Q8 group has the same order as the dihedral group DiH4, 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.
The Cayley table (multiplication table) for Q is given by:
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.
The quaternion group is the smallest example of nilpotent non-abelian group. Another example of nilpotent non-abelian group is dihedral group of order eight (generated by an element of order four and an element of order two that conjugates the element of order four to its inverse).
The quaternion group has five irreducible representations, and their dimensions are 1,1,1,1,2. The proof for this property is not hard, since the number of irreducible characters of the quaternion group is equal to the number of conjugacy classes of the quaternion group, which is five ( { e }, { e }, { i, i }, { j, j }, { k, k } ).
These five representations are:
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