quaternion group

The quaternion groupMathworldPlanetmathPlanetmath, or quaternionic group, is a noncommutative groupMathworldPlanetmath with eight elements. It is traditionally denoted by Q (not to be confused with ) or by Q8. This group is defined by the presentationMathworldPlanetmathPlanetmathPlanetmath


or, equivalently, defined by the multiplication table

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 subgroupMathworldPlanetmathPlanetmath contained in the center of Q. Every subgroup of Q is normal and, except for the trivial subgroup {1}, contains {1,-1}. The dihedral groupMathworldPlanetmath D4 (the group of symmetries of a square) is the only other noncommutative group of order 8.

Since i2=j2=k2=-1, the elements i, j, and k are known as the imaginary unitsMathworldPlanetmath, by analogy with i. Any pair of the imaginary units generate the group. Better, given x,y{i,j,k}, any element of Q is expressible in the form xmyn.

Q is identified with the group of units (invertible elements) of the ring of quaternionsMathworldPlanetmath over . That ring is not identical to the group ringMathworldPlanetmath [Q], which has dimension 8 (not 4) over . Likewise the usual quaternion algebra is not quite the same thing as the group algebra [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.

Title quaternion group
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