Math 2.0

Group theory and generalizations Example

The Klein four-group is abelian but not cyclic

Builds on Klein four-groupOrder of an element

Example

The Klein four-group is V={e,a,b,c}V = \{e, a, b, c\}, where a2=b2=c2=ea^2 = b^2 = c^2 = e and ab=ba=cab = ba = c. The group is abelian. Each element that is not ee has order 22. Thus each element generates a subgroup with at most 22 elements. No element generates all 44 elements, so VV is not cyclic.

Remarks

Thus the converse of the result that cyclic groups are abelian is false. Up to isomorphism, VV and ℤ/4ℤ\mathbb{Z}/4\mathbb{Z} are the only groups of order 44.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph
Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionDirect product of groups, definition, no evidence levelDirect product of gro…definitionLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementAbelian group, definition, no evidence levelAbelian groupdefinitionOrder of an element, definition, no evidence levelOrder of an elementdefinitionKlein four-group, object, no evidence levelKlein four-groupobjectThe Klein four-group is abelian but not cyclic, example, no evidence levelThe Klein four-group …example
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. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Direct product of groups definition
  5. Laws of exponents in a group statement
  6. Abelian group definition
  7. Order of an element definition
  8. Klein four-group object
  9. The Klein four-group is abelian but not cyclic example

Further reading

  1. Web https://en.wikipedia.org/wiki/Klein_four-group, lead section
About this page
TagDX6X1D21FE
URIm2://DX6X1D21FE
Kindexample
WritingW0 stub
MSC20A05
Instance ofAbelian group
Not an instance ofCyclic group (note: no element generates the group.), Cyclic group (note: each element generates a subgroup with at most two elements.)
UsesKlein four-group, Order of an element

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

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

What links here

No page links to this page yet.