Quaternion group

In group theory, the quaternion group is a non-abelian group of order 8 with a number of interesting properties. It is often given the symbol Q₈.

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

Q₈ = {1, -1, i, -i, j, -j, k, -k}.

Here 1 is the identity element, (-1) · (-1) = 1, and -1 · a = a · (-1) for all a; we write this latter element as -a. The remaining relations can be obtained from the following multiplication table:

	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.

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

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}.

Q₈ has a presentation with generators {x,y} and relations x⁴ = 1, x² = y², and y^-1xy = 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 x²ⁿ = 1, x^2n-1 = y², and y^-1xy = x^-1. These groups are members of the still larger family of dicyclic groups.