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Concept
Klein four-group
Summary
In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one.
It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal reflection, vertical reflection and 180-degree rotation),
as the group of bitwise exclusive or operations on two-bit binary values,
or more abstractly as Z2 × Z2, the direct product of two copies of the cyclic group of order 2.
It was named Vierergruppe (meaning four-group) by Felix Klein in 1884.
It is also called the Klein group, and is often symbolized by the letter V or as K4.
The Klein four-group, with four elements, is the smallest group that is not a cyclic group. There is only one other group of order four, up to isomorphism, the cyclic group of order 4. Both are abelian groups.
The Klein group's Cayley table is given by:
The Klein four-group is also defined by the group presentation
All non-identity elements of the Klein group have order 2, thus any two non-identity elements can serve as generators in the above presentation. The Klein four-group is the smallest non-cyclic group. It is however an abelian group, and isomorphic to the dihedral group of order (cardinality) 4, i.e. D4 (or D2, using the geometric convention); other than the group of order 2, it is the only dihedral group that is abelian.
The Klein four-group is also isomorphic to the direct sum Z2 ⊕ Z2, so that it can be represented as the pairs {(0,0), (0,1), (1,0), (1,1)} under component-wise addition modulo 2 (or equivalently the bit strings {00, 01, 10, 11} under bitwise XOR); with (0,0) being the group's identity element. The Klein four-group is thus an example of an elementary abelian 2-group, which is also called a Boolean group. The Klein four-group is thus also the group generated by the symmetric difference as the binary operation on the subsets of a powerset of a set with two elements, i.
Official source
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