Quaternion group
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GroupDiagramQ8.png
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.