# quaternion group

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^{}

$$\{i,j;{i}^{4},{i}^{2}{j}^{2},ij{i}^{-1}j\}$$ |

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 $i\in \u2102$. 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 ${x}^{m}{y}^{n}$.

$Q$ 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.

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 |