# Quaternion group

where *e* is the identity element and *e* commutes with the other elements of the group.

The quaternion group Q_{8} has the same order as the dihedral group D_{4}, but a different structure, as shown by their Cayley and cycle graphs:

In the diagrams for D_{4}, the group elements are marked with their action on a letter F in the defining representation **R**^{2}. The same cannot be done for Q_{8}, since it has no faithful representation in **R**^{2} or **R**^{3}. D_{4} can be realized as a subset of the split-quaternions in the same way that Q_{8} can be viewed as a subset of the quaternions.

The elements *i*, *j*, and *k* all have order four in Q_{8} and any two of them generate the entire group. Another presentation of Q_{8}^{[2]} based in only two elements to skip this redundancy is:

The quaternion group has the unusual property of being Hamiltonian: Q_{8} is non-abelian, but every subgroup is normal.^{[3]} Every Hamiltonian group contains a copy of Q_{8}.^{[4]}

The quaternion group Q_{8} and the dihedral group D_{4} are the two smallest examples of a nilpotent non-abelian group.

**Sign representations with i,j,k-kernel**: Q_{8} 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 Q_{8} in the special linear group SL(2,**C**).^{[5]}

Also, this representation permutes the eight non-zero vectors of (**F**_{3})^{2}, giving an embedding of Q_{8} in the symmetric group S_{8,} 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** Q_{4n} of order 4*n* 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 SL_{2}(*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 *p ^{r}* be the size of

*F*, where

*p*is prime, the size of the 2-Sylow subgroup of SL

_{2}(

*F*) is 2

^{n}, where

*n*= ord

_{2}(

*p*

^{2}− 1) + ord

_{2}(

*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