In mathematics , the **Klein group** or the **Vierergruppe** (in German, *group of four* ) is the abelian group of four elements in which each element is the inverse of itself. It receives this name in honor of the German Félix Klein .

Let C _{2} = {1,2} be the cyclic group of order two; that is: 2 ^{2} = 1. Also let H = C _{2} xC _{2} = {(1, 1), (1, 2), (2, 1), (2, 2)}. Then the product of ordered pairs in C _{2} xC _{2 can be performed} . For example (1, 2)) x (2, 1) = (1×2, 2×1) = (2,2). Other cases: (2, 2) x (2, 2) = (1,1) ^{[1]} It is observed that the product of any element by itself is the unit = (1,1), so there is no generator in H. So H is not isomorphic to the cyclic group C _{4} = {1, b, b ^{2} , b ^{3} } of order 4. We also have (1; 1) ^{2} = (1; 1), (1, 2) ^{2} = (1,1), (2,1) ^{2}= (1,1) and (2; 2) ^{2} = (1,1), the order of each element is 2, except for the identity = (1,1), just as each element is inverse of itself.

- With the identification 1 = (1,2); a = (1,2); b = (2,1) and c = (2,2) the table that appears in the following section is generated.

## Summary

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- 1 Group definition
- 2 Algebraic properties
- 3 See also
- 4 References and notes

## Group definition

The Klein group is the group *(K, •)* where *K = {1, a, b, c}* and whose internal binary operation **•** is defined by the following table:

• |
one |
to |
b |
c |

one |
one | to | b | c |

to |
to | one | c | b |

b |
b | c | one | to |

c |
c | b | to | one |

Sometimes, since *c = a • b* , we write *K = {1, a, b, ab}* . With these names, the operation table is

• |
one |
to |
b |
ab |

one |
one | to | b | ab |

to |
to | one | ab | b |

b |
b | ab | one | to |

ab |
ab | b | to | one |

## Algebraic properties

- The Klein group is isomorphic to the direct productof the cyclic group of order 2 by itself:
*K = Z*_{2}*x Z*._{2} - Klein’s group is abelian
^{[2]}, that is, the internal operation is commutative. - The Klein group is not cyclical, so there is no generating element and, therefore, it is not isomorphic to the
*Z*group ._{2} - The order of all elements is 2, except for the neutral element 1 whose order is 1.
^{[3]} - Klein’s group has the representation
*<a, b | a*^{2}*= b*^{2}*= (ab)*^{2}*= 1>*^{[4]} - Klein’s group only has three of its own subgroups, isomorphic to the cyclic group of order 2. They are generated by each of the elements other than neutral 1.
- The group of rhombus turns is isomorphic to the Klein group
^{[5]} - There is an alternating subgroup of order 4, from the symmetric group S
_{4}that is isomorphic with the Klein group.