# Element structure of Klein four-group

## Contents |

This article gives specific information, namely, element structure, about a particular group, namely: Klein four-group.

View element structure of particular groups | View other specific information about Klein four-group

The Klein four-group is the unique (up to isomorphism) non-cyclic group of order four. In this group, every non-identity element has order two.

The multiplication table with non-identity elements and identity element :

Element/element | ||||
---|---|---|---|---|

The multiplication table can be described as follows (and this characterizes the group):

- The product of the identity element and any element is that element itself.
- The product of any non-identity element with itself is the identity element.
- The product of two distinct non-identity elements is the third non-identity element.

## Elements

Below is a description of the elements for the many alternate descriptions of the Klein four-group. Note that the choice of correspondence between the descriptions is somewhat arbitrary, in the sense that it can be modified by automorphisms of the Klein four-group, which include arbitrary permutations of the three non-identity elements (see endomorphism structure of Klein four-group).

Element | Interpretation as , viewing as the additive group of integers mod 2 and denotes external direct product | Interperation as | Interpretation as double transposition |
---|---|---|---|

(0,0) | (1,1) | ||

(1,0) | (-1,1) | ||

(0,1) | (1,-1) | ||

(1,1) | (-1,-1) |

Here is the multiplication table (better termed an *addition table*, since we are carrying out coordinate-wise addition mod 2) viewed as a direct product of two copies of the additive group of integers mod 2:

Element/element | (0,0) | (1,0) | (0,1) | (1,1) |
---|---|---|---|---|

(0,0) | (0,0) | (1,0) | (0,1) | (1,1) |

(1,0) | (1,0) | (0,0) | (1,1) | (0,1) |

(0,1) | (0,1) | (1,1) | (0,0) | (1,0) |

(1,1) | (1,1) | (0,1) | (1,0) | (0,0) |

Here is the multiplication table viewed as a direct product of two copies of :

Element/element | (1,1) | (-1,1) | (1,-1) | (-1,-1) |
---|---|---|---|---|

(1,1) | (1,1) | (-1,1) | (1,-1) | (-1,-1) |

(-1,1) | (-1,1) | (1,1) | (-1,-1) | (1,-1) |

(1,-1) | (1,-1) | (-1,-1) | (1,1) | (-1,1) |

(-1,-1) | (-1,-1) | (1,-1) | (-1,1) | (1,1) |