Quaternion group

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Missing image
GroupDiagramQ8.png
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 8 with a number of interesting properties. The quaternion group, often denoted by Q, is usually written in multiplicative form, with the following 8 elements

Q = {1, −1, i, −i, j, −j, k, −k}

Here 1 is the identity element, (−1)2 = 1, and (−1)a = a(−1) = −a for all a in Q. The remaining multiplication rules can be obtained from the following relation:

<math>i^2 = j^2 = k^2 = ijk = -1<math>

The entire multiplication table for Q is given by:

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

Note that the resulting group is non-commutative; for example ij = −ji. Q 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 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. Note that this is not quite the group algebra on Q (which would be 8-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}.

Note that i, j, and k all have order 4 in Q and any two of them generate the entire group. Q has the presentation

<math>\langle x,y \mid x^4 = 1, x^2 = y^2, yxy^{-1} = x^{-1}\rangle<math>

One may take, for instance, x = i and y = j.

The center 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.

Generalized quaternion group

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

<math>\langle x,y \mid x^{2^{n-1}} = 1, x^{2^{n-2}} = y^2, yxy^{-1} = x^{-1}\rangle<math>

for some integer n ≥ 3. The order of this group is 2n. The ordinary quaternion group corresponds to the case n = 3. The generalized quaternion group can be realized as the subgroup of unit quaternions generated by

<math>x = e^{2\pi i/2^{n-1}}<math>
<math>y = j\,<math>

The generalized quaternion groups are members of the still larger family of dicyclic groups.

See also

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