In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset of the quaternions under multiplication. It is given by the group presentation
where e is the identity element and commutes with the other elements of the group.
Another presentation of Q8 is
The quaternion group sprang full-blown from the mind of W. R. Hamilton, and there has been an effort to connect it with the wellspring of discrete groups in field extensions and the study of algebraic numbers. Richard Dedekind considered the field ℚ[√2, √3] in this effort. In field theory, extensions are generated by roots of polynomial irreducible over a ground field. Isomorphic fields are associated with permutation groups that move around the roots of the polynomial. This notion, pioneered by Évariste Galois (1830), now forms a standard study Galois theory in mathematics education. In 1936 Ernst Witt published his approach to the quaternion group through Galois theory. A definitive connection was published in 1981, see § Galois group.
Compared to dihedral 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:
In the diagrams for D4, the group elements are marked with their action on a letter F in the defining representation R2. The same cannot be done for Q8, since it has no faithful representation in R2 or R3. 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.
Cayley table
The Cayley table (multiplication table) for Q8 is given by:
Properties
The elements i, j, and k all have order four in Q8 and any two of them generate the entire group. Another presentation of Q8 based in only two elements to skip this redundancy is:
One may take, for instance, and .
The quaternion group has the unusual property of being Hamiltonian: Q8 is non-abelian, but every subgroup is normal. Every Hamiltonian group contains a copy of Q8.
The quaternion group Q8 and the dihedral group D4 are the two smallest examples of a nilpotent non-abelian group.
The center and the commutator subgroup of Q8 is the subgroup . The inner automorphism group of Q8 is given by the group modulo its center, i.e. the factor group which is isomorphic to the Klein four-group V. The full automorphism group of Q8 is isomorphic to S4, the symmetric group on four letters (see Matrix representations below), and the outer automorphism group of Q8 is thus S4/V, which is isomorphic to S3.
The quaternion group Q8 has five conjugacy classes, and so five irreducible representations over the complex numbers, with dimensions 1, 1, 1, 1, 2:
Trivial representation.
Sign representations with i, j, k-kernel: Q8 has three maximal normal subgroups: the cyclic subgroups generated by i, j, and k respectively. For each maximal normal subgroup N, we obtain a one-dimensional representation factoring through the 2-element quotient group G/N. The representation sends elements of N to 1, and elements outside N to −1.
2-dimensional representation: Described below in Matrix representations.
The character table of Q8 turns out to be the same as that of D4:
Since the irreducible characters in the rows above have real values, this gives the decomposition of the real group algebra of into minimal two-sided ideals:
where the idempotents correspond to the irreducibles:
so that
Each of these irreducible ideals is isomorphic to a real central simple algebra, the first four to the real field . The last ideal is isomorphic to the skew field of quaternions by the correspondence:
Furthermore, the projection homomorphism given by has kernel ideal generated by the idempotent:
so the quaternions can also be obtained as the quotient ring .
The complex group algebra is thus where is the algebra of biquaternions.
Matrix representations
The two-dimensional irreducible complex representation described above gives the quaternion group Q8 as a subgroup of the general linear group . The quaternion group is a multiplicative subgroup of the quaternion algebra:
which has a regular representation by left multiplication on itself considered as a complex vector space with basis so that corresponds to the -linear mapping The resulting representation
is given by:
Since all of the above matrices have unit determinant, this is a representation of Q8 in the special linear group .
A variant gives a representation by unitary matrices (table at right). Let correspond to the linear mapping so that is given by: