The quaternion group is usually written in multiplicative form, with the following 8 elements

- Q
_{8}= {1, -1,*i*, -*i*,*j*, -*j*,*k*, -*k*}.

i | j | k | |

i | -1 | k | -j |

j | -k | -1 | i |

k | j | -i | -1 |

Note that the resulting group is non-commutative; for example *ij* = -*ji*.

*Q*_{8} has the unusual property of being Hamiltonian: every subgroup of *Q*_{8} is a normal subgroup, but the group is non-abelian. Every Hamiltonian group contains a copy of *Q*_{8}.

In abstract algebra, we can construct a real 4-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.

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*}.

*Q*_{8} has a presentation with generators {*x*,*y*} and relations *x*^{4} = 1, *x*^{2} = *y*^{2}, and *y*^{-1}*xy* = *x*^{-1}. (For example *x* = *i*, *y* = *j*.) A group is called a **generalized quaternion group** if it has a presentation, for some integer *n* > 1, with generators {*x*,*y*} and relations *x*^{2n} = 1, *x*^{2n-1} = *y*^{2}, and *y*^{-1}*xy* = *x*^{-1}. These groups are members of the still larger family of dicyclic groups.