Klein four-group
In mathematics, the Klein four-group (or just Klein group or Vierergruppe (English: four-group), often symbolized by the letter V or as K4) is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe by Felix Klein in 1884.
With four elements, the Klein four-group is the smallest non-cyclic group, and the cyclic group of order 4 and the Klein four-group are, up to isomorphism, the only groups of order 4. Both are abelian groups. The smallest non-abelian group is the symmetric group of degree 3, which has order 6.
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, Dih2; 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.e. over a field of sets with four elements, e.g. ; the empty set is the group's identity element in this case.
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