Quaternion group

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:

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.

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[2] based in only two elements to skip this redundancy is:

The quaternion group has the unusual property of being Hamiltonian: Q8 is non-abelian, but every subgroup is normal.[3] Every Hamiltonian group contains a copy of Q8.[4]

The quaternion group Q8 and the dihedral group D4 are the two smallest examples of a nilpotent non-abelian group.

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.

Since all of the above matrices have unit determinant, this is a representation of Q8 in the special linear group SL(2,C).[5]

Multiplication table of the quaternion group as a subgroup of SL(2,3). The field elements are denoted 0,+,−.

Also, this representation permutes the eight non-zero vectors of (F3)2, giving an embedding of Q8 in the symmetric group S8, in addition to the embeddings given by the regular representations.

As Richard Dean showed in 1981, the quaternion group can be presented as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field over Q 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.[6]

A generalized quaternion group Q4n of order 4n is defined by the presentation[2]

The generalized quaternion groups have the property that every abelian subgroup is cyclic.[9] 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.[10] 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.[11] In particular, for a finite field F with odd characteristic, the 2-Sylow subgroup of SL2(F) is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group, (Gorenstein 1980, p. 42). Letting pr be the size of F, where p is prime, the size of the 2-Sylow subgroup of SL2(F) is 2n, where n = ord2(p2 − 1) + ord2(r).

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,[12] which admits the presentation