quaternion group
The quaternion group, or quaternionic group, is a noncommutative group with eight elements. It is traditionally denoted by (not to be confused with ) or by . This group is defined by the presentation
or, equivalently, defined by the multiplication table
where we have put each product into row and column . The minus signs are justified by the fact that is subgroup contained in the center of . Every subgroup of is normal and, except for the trivial subgroup , contains . The dihedral group (the group of symmetries of a square) is the only other noncommutative group of order 8.
Since , the elements , , and are known as the imaginary units, by analogy with . Any pair of the imaginary units generate the group. Better, given , any element of is expressible in the form .
is identified with the group of units (invertible elements) of the ring of quaternions over . That ring is not identical to the group ring , which has dimension 8 (not 4) over . Likewise the usual quaternion algebra is not quite the same thing as the group algebra .
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.
Title | quaternion group |
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Canonical name | QuaternionGroup |
Date of creation | 2013-03-22 12:35:35 |
Last modified on | 2013-03-22 12:35:35 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 12 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 20A99 |
Synonym | quaternionic group |
Related topic | Quaternions |
Defines | quaternion group |