Math 2.0

Group theory and generalizations Finite groups Object

Klein four-group

Builds on Abelian groupDirect product of groups

Description

The Klein four-group VV is the group with four elements e,a,b,ce, a, b, c, in which ee is the identity and each element is its own inverse. The product of two distinct elements of {a,b,c}\{a, b, c\} is the third one. It is abelian, and it is isomorphic to the direct product ℤ/2ℤ×ℤ/2ℤ\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}.

Remarks

The Klein four-group is the smallest group that is not cyclic, because it has no element of order 44. It is the symmetry group of a rectangle that is not a square. In S4S_4, the set {e,(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)}\{e, (1\ 2)(3\ 4), (1\ 3)(2\ 4), (1\ 4)(2\ 3)\} is a normal subgroup isomorphic to VV. Each group of order 44 is isomorphic to ℤ/4ℤ\mathbb{Z}/4\mathbb{Z} or to VV.

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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…definitionAbelian group, definition, no evidence levelAbelian groupdefinitionKlein four-group, object, no evidence levelKlein four-groupobject
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  1. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Direct product of groups definition
  5. Abelian group definition
  6. Klein four-group object

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3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Klein_four-group, lead section
  2. Web https://en.wikipedia.org/wiki/Klein_four-group, Permutation representation
About this page
Tag0PMRQJVTME
URIm2://0PMRQJVTME
Aliasesvierergruppe
Kindobject
WritingW1 stated
MSC20K01
UsesAbelian group, Direct product of groups
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