Klein four-group

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

Z

2

×

Z

2

{\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}

, the direct product of two copies of the cyclic group of order 2 by the Fundamental Theorem of Finitely Generated Abelian Groups. It was named Vierergruppe (German: [ˈfiːʁɐˌɡʁʊpə] , meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter

V

{\displaystyle V}

or as

K

4

{\displaystyle K_{4}}

.

The Klein four-group, with four elements, is the smallest group that is not cyclic. Up to isomorphism, there is only one other group of order four: the cyclic group of order 4. Both groups are abelian.

Presentations

The Klein group's Cayley table is given by:

The Klein four-group is also defined by the group presentation

V

=

⟨

a

,

b

∣

a

2

=

b

2

=

(

a

b

)

2

=

e

⟩

.

{\displaystyle V=\left\langle a,b\mid a^{2}=b^{2}=(ab)^{2}=e\right\rangle .}

All non-identity elements of the Klein group have order 2, so any two non-identity elements can serve as generators in the above presentation. The Klein four-group is the smallest non-cyclic group.