# Quaternion group Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). For example, the cycle in red reflects the fact that i 2 = −1, i 3 = −i  and i 4 = 1. The red cycle also reflects the fact that (−i )2 = −1, (−i )3 = i  and (−i )4 = 1.

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

$\mathrm {Q} =\langle -1,i,j,k\mid (-1)^{2}=1,\;i^{2}=j^{2}=k^{2}=ijk=-1\rangle ,\,\!$ where 1 is the identity element and −1 commutes with the other elements of the group.

## Graphs

The Q8 group has the same order as the Dihedral group, D4, but a different structure, as shown by their Cayley graphs:

## Cayley table

The Cayley table (multiplication table) for Q is given by:

1 −1 i −i j −j k −k
1 1 −1 i −i j −j k −k
−1 −1 1 −i i −j j −k k
i i −i −1 1 k −k −j j
−i −i i 1 −1 −k k j −j
j j −j −k k −1 1 i −i
−j −j j k −k 1 −1 −i i
k k −k j −j −i i −1 1
−k −k k −j j i −i 1 −1

The multiplication of pairs of elements from the subset {±i, ±j, ±k} works like the cross product of unit vectors in three-dimensional Euclidean space.

{\begin{alignedat}{2}ij&=k,&ji&=-k,\\jk&=i,&kj&=-i,\\ki&=j,&ik&=-j.\end{alignedat}} ## Properties

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.

In abstract algebra, one can construct a real four-dimensional vector space with basis {1, i, j, k} and turn it into an associative algebra by using the above multiplication table and distributivity. The result is a skew field called the quaternions. Note that this is not quite the same as the group algebra on Q (which would be eight-dimensional). Conversely, one can start with the quaternions and define the quaternion group as the multiplicative subgroup consisting of the eight elements {1, −1, i, −i, j, −j, k, −k}. The complex four-dimensional vector space on the same basis is called the algebra of biquaternions.

Note that i, j, and k all have order four in Q and any two of them generate the entire group. Another presentation of Q demonstrating this is:

$\langle x,y\mid x^{4}=1,x^{2}=y^{2},y^{-1}xy=x^{-1}\rangle .\,\!$ One may take, for instance, i = x, j = y and k = x y.

The center and the commutator subgroup of Q is the subgroup {±1}. The factor group Q/{±1} is isomorphic to the Klein four-group V. The inner automorphism group of Q is isomorphic to Q modulo its center, and is therefore also isomorphic to the Klein four-group. The full automorphism group of Q is isomorphic to S4, the symmetric group on four letters. The outer automorphism group of Q is then S4/V which is isomorphic to S3.

## Matrix representations

The quaternion group can be represented as a subgroup of the general linear group GL2(C). A representation

$\mathrm {Q} =\{\pm 1,\pm i,\pm j,\pm k\}\to \mathrm {GL} _{2}(\mathbf {C} )$ is given by

$1\mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}}$ $i\mapsto {\begin{pmatrix}i&0\\0&-i\end{pmatrix}}$ $j\mapsto {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}$ $k\mapsto {\begin{pmatrix}0&i\\i&0\end{pmatrix}}$ Since all of the above matrices have unit determinant, this is a representation of Q in the special linear group SL2(C). The standard identities for quaternion multiplication can be verified using the usual laws of matrix multiplication in GL2(C).

There is also an important action of Q on the eight nonzero elements of the 2-dimensional vector space over the finite field F3. A representation

$\mathrm {Q} =\{\pm 1,\pm i,\pm j,\pm k\}\to \mathrm {GL} (2,3)$ is given by

$1\mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}}$ $i\mapsto {\begin{pmatrix}1&1\\1&-1\end{pmatrix}}$ $j\mapsto {\begin{pmatrix}-1&1\\1&1\end{pmatrix}}$ $k\mapsto {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}$ where {−1,0,1} are the three elements of F3. Since all of the above matrices have unit determinant over F3, this is a representation of Q in the special linear group SL(2, 3). Indeed, the group SL(2, 3) has order 24, and Q is a normal subgroup of SL(2, 3) of index 3.

## Galois group

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

$x^{8}-72x^{6}+180x^{4}-144x^{2}+36$ .

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.

## Generalized quaternion group

A group is called a generalized quaternion group or dicyclic group if it has a presentation

$\langle x,y\mid x^{2n}=y^{4}=1,x^{n}=y^{2},y^{-1}xy=x^{-1}\rangle .\,\!$ for some integer n ≥ 2. This group is denoted Q4n and has order 4n. Coxeter labels these dicyclic groups <2,2,n>, being a special case of the binary polyhedral group <l,m,n> and related to the polyhedral groups (p,q,r), and dihedral group (2,2,n). The usual quaternion group corresponds to the case n = 2. The generalized quaternion group can be realized as the subgroup of GL2(C) generated by

$\left({\begin{array}{cc}\omega _{n}&0\\0&{\overline {\omega }}_{n}\end{array}}\right){\mbox{ and }}\left({\begin{array}{cc}0&-1\\1&0\end{array}}\right)$ where ωn = eiπ/n. It can also be realized as the subgroup of unit quaternions generated by x = eiπ/n and y = j.

The generalized quaternion groups have the property that every abelian subgroup is cyclic. 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. Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or generalized quaternion (of order a power of 2). 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, Template:Harv. 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 groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.