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.
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.
|Date of creation||2013-03-22 12:35:35|
|Last modified on||2013-03-22 12:35:35|
|Last modified by||mathcam (2727)|