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quaternion group (Definition)

The quaternion group, or quaternionic group, is a noncommutative group with eight elements. It is traditionally denoted by $ Q$ (not to be confused with $ \mathbb{Q}$) or by $ Q_8$. This group is defined by the presentation

$\displaystyle \{i,j;i^4,i^2j^2,iji^{-1}j\}$
or, equivalently, defined by the multiplication table
$ \cdot$ $ 1$ $ i$ $ j$ $ k$ $ -i$ $ -j$ $ -k$ $ -1$
$ 1$ $ 1$ $ i$ $ j$ $ k$ $ -i$ $ -j$ $ -k$ $ -1$
$ i$ $ i$ $ -1$ $ k$ $ -j$ $ 1$ $ -k$ $ j$ $ -i$
$ j$ $ j$ $ -k$ $ -1$ $ i$ $ k$ $ 1$ $ -i$ $ -j$
$ k$ $ k$ $ j$ $ -i$ $ -1$ $ -j$ $ i$ $ 1$ $ -k$
$ -i$ $ -i$ $ 1$ $ -k$ $ j$ $ -1$ $ k$ $ -j$ $ i$
$ -j$ $ -j$ $ k$ $ 1$ $ -i$ $ -k$ $ -1$ $ i$ $ j$
$ -k$ $ -k$ $ -j$ $ i$ $ 1$ $ j$ $ -i$ $ -1$ $ k$
$ -1$ $ -1$ $ -i$ $ -j$ $ -k$ $ i$ $ j$ $ k$ $ 1$

where we have put each product $ xy$ into row $ x$ and column $ y$. The minus signs are justified by the fact that $ \{1,-1\}$ is subgroup contained in the center of $ Q$. Every subgroup of $ Q$ is normal and, except for the trivial subgroup $ \{1\}$, contains $ \{1,-1\}$. The dihedral group $ D_4$ (the group of symmetries of a square) is the only other noncommutative group of order 8.

Since $ i^2 = j^2 = k^2 = -1$, the elements $ i$, $ j$, and $ k$ are known as the imaginary units, by analogy with $ i\in\mathbb{C}$. Any pair of the imaginary units generate the group. Better, given $ x,y\in\{i,j,k\}$, any element of $ Q$ is expressible in the form $ x^my^n$.

$ Q$ is identified with the group of units (invertible elements) of the ring of quaternions over $ \mathbb{Z}$. That ring is not identical to the group ring $ \mathbb{Z}[Q]$, which has dimension 8 (not 4) over $ \mathbb{Z}$. Likewise the usual quaternion algebra is not quite the same thing as the group algebra $ \mathbb{R}[Q]$.

Quaternions were known to Gauss in 1819 or 1820, but he did not publicize this discovery, and quaternions weren't rediscovered until 1843, with Hamilton.



"quaternion group" is owned by mathcam. [ full author list (4) | owner history (3) ]
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See Also: Hamiltonian quaternions

Other names:  quaternionic group
Also defines:  quaternion group

Attachments:
generalized quaternion group (Derivation) by Algeboy
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Cross-references: Gauss, group algebra, quaternion algebra, dimension, group ring, quaternions, ring, invertible, group of units, expressible, generate, imaginary units, square, symmetries, dihedral group, contains, trivial subgroup, normal, center, contained, subgroup, column, row, product, multiplication, presentation, group, noncommutative group
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This is version 9 of quaternion group, born on 2002-04-17, modified 2006-07-07.
Object id is 2844, canonical name is QuaternionGroup.
Accessed 8823 times total.

Classification:
AMS MSC: 20A99 (Group theory and generalizations :: Foundations :: Miscellaneous)
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