Quaternion group

These relations, discovered by W. R. Hamilton, also generate the quaternions as an algebra over the real numbers.

[1] 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.

Another presentation of Q8[3] based in only two elements to skip this redundancy is:

For instance, writing the group elements in lexicographically minimal normal forms, one may identify:

The quaternion group has the unusual property of being Hamiltonian: Q8 is non-abelian, but every subgroup is normal.

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.

For each maximal normal subgroup N, we obtain a one-dimensional representation factoring through the 2-element quotient group G/N.

It is not realizable over the real numbers, but is a complex representation: indeed, it is just the quaternions

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

has kernel ideal generated by the idempotent: so the quaternions can also be obtained as the quotient ring

, but splits into two copies of the two-dimensional irreducible when extended to the complex numbers.

The two-dimensional irreducible complex representation described above gives the quaternion group Q8 as a subgroup of the general linear group

by left multiplication on itself considered as a complex vector space with basis

[6] A variant gives a representation by unitary matrices (table at right).

is given by: It is worth noting that physicists exclusively use a different convention for the

matrix representation to make contact with the usual Pauli matrices: This particular choice is convenient and elegant when one describes spin-1/2 states in the

basis and considers angular momentum ladder operators

There is also an important action of Q8 on the 2-dimensional vector space over the finite field

admits a linear mapping: In addition we have the Frobenius automorphism

in attempting to relate the quaternion group to Galois theory.

[7] In 1936 Ernst Witt published his approach to the quaternion group through Galois theory.

[8] In 1981, Richard Dean showed the quaternion group can be realized as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field of the polynomial The development uses the fundamental theorem of Galois theory in specifying four intermediate fields between Q and T and their Galois groups, as well as two theorems on cyclic extension of degree four over a field.

, a special case of the binary polyhedral group

The generalized quaternion group can be realized as the subgroup of

[3] It can also be realized as the subgroup of unit quaternions generated by[10]

The generalized quaternion groups have the property that every abelian subgroup is cyclic.

[11] It can be shown that a finite p-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above.

[12] Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or a 2-group isomorphic to generalized quaternion group.

The Brauer–Suzuki theorem shows that the groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.

Another terminology reserves the name "generalized quaternion group" for a dicyclic group of order a power of 2,[14] which admits the presentation