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:

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 xyx = y. (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, xn = y2, and xyx = y. These groups are members of the still larger family of dicyclic groups.}