Editing Klein four-group (section)

== Presentations ==
The Klein group's [[Cayley table]] is given by:
{| class=wikitable width=120
!*
!e
!''a''
!''b''
!''c''
|- align=center
!e
|e || ''a'' || ''b'' || ''c''
|- align=center
!''a''
| ''a'' || e || ''c'' || ''b''
|- align=center
!''b''
|''b'' || ''c'' || e || ''a''
|- align=center
!''c''
| ''c'' || ''b'' || ''a'' || e
|}

The Klein four-group is also defined by the [[presentation of a group|group presentation]]
: <math>V = \left\langle a,b \mid a^2 = b^2 = (ab)^2 = e \right\rangle.</math>

All non-[[identity element|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]]. It is, however, an [[abelian group]], and isomorphic to the [[dihedral group]] of order (cardinality) 4, symbolized <math>D_4</math> (or <math>D_2</math>, 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]] <math>\mathbb{Z}_2\oplus\mathbb{Z}_2</math>, so that it can be represented as the pairs {{nowrap|{(0,0), (0,1), (1,0), (1,1)} }} under component-wise addition [[Modular arithmetic|modulo 2]] (or equivalently the [[Bit array|bit strings]] {{nowrap|{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 group|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 [[subset]]s of a [[powerset]] of a set with two elements—that is, over a [[field of sets]] with four elements, such as <math>\{\emptyset,\{\alpha\},\{\beta\},\{\alpha,\beta\}\}</math>; the [[empty set]] is the group's identity element in this case.

Another numerical construction of the Klein four-group is the set {{nowrap|{{mset| 1, 3, 5, 7 }}}}, with the operation being [[Multiplicative group of integers modulo n|multiplication modulo 8]]. Here ''a'' is 3, ''b'' is 5, and {{nowrap|1=''c'' = ''ab''}} is {{nowrap|1=3 × 5 = 15 ≡ 7 (mod 8)}}.

The Klein four-group also has a representation as {{nowrap|2 × 2}} real matrices with the operation being matrix multiplication:
: <math>
  e =\begin{pmatrix}
      1 & 0\\
      0 & 1
    \end{pmatrix},\quad
  a = \begin{pmatrix}
      1 &  0\\
      0 & -1
    \end{pmatrix},\quad
</math>
: <math>
  b = \begin{pmatrix}
      -1 & 0\\
       0 & 1
    \end{pmatrix},\quad
  c = \begin{pmatrix}
      -1 &  0\\
       0 & -1
    \end{pmatrix}
</math>

On a [[Rubik's Cube]], the "4 dots" pattern can be made in three ways (for example, M2 U2 M2 U2 F2 M2 F2), depending on the pair of faces that are left blank; these three positions together with the solved position form an example of the Klein group, with the solved position serving as the identity.