Math 2.0

Group theory and generalizations Finite groups Object

Quaternion group

Builds on GroupCentre of a group

Description

The quaternion group Q8Q_8 is the set {1,−1,i,−i,j,−j,k,−k}\{1, -1, i, -i, j, -j, k, -k\} of quaternions, with quaternion multiplication as the operation. The multiplication follows from the rules i2=j2=k2=ijk=−1i^2 = j^2 = k^2 = ijk = -1, where −1-1 commutes with each element. These rules give ij=kij = k, jk=ijk = i, ki=jki = j and ji=−kji = -k, kj=−ikj = -i, ik=−jik = -j. The group has order 88, and it is not abelian.

Remarks

The element −1-1 is the only element of order 22, and the six elements ±i,±j,±k\pm i, \pm j, \pm k have order 44. The centre and the commutator subgroup of Q8Q_8 are both {1,−1}\{1, -1\}, and Q8/{1,−1}Q_8 / \{1, -1\} is the Klein four-group. Each subgroup of Q8Q_8 is normal, although Q8Q_8 is not abelian. The groups Q8Q_8 and D4D_4 are not isomorphic, because D4D_4 has five elements of order 22.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptCentre of a group, definition, no evidence levelCentre of a groupdefinitionQuaternion group, object, no evidence levelQuaternion groupobject
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Centre of a group definition
  3. Quaternion group object

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quaternion_group, lead section
  2. Web https://en.wikipedia.org/wiki/Quaternion_group, Properties
  3. Web https://en.wikipedia.org/wiki/Quaternion_group, Compared to dihedral group
About this page
Tag0NRJGW864D
URIm2://0NRJGW864D
Aliasesq8
Kindobject
WritingW1 stated
MSC20D15
UsesGroup, Centre of a group
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • section Invariants: a table of the main invariants (0 of 1)
  • edge instance_of out (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by