# Z2 in V4

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z2 and the group is (up to isomorphism) Klein four-group (see subgroup structure of Klein four-group).

The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part| Group-subgroup pairs with the same quotient part | All pages on particular subgroups in groups

## Contents

## Definition

We are interested in the Klein four-group:

where is the identity element and are all non-identity elements of order 2.

We are interested in the subgroup:

as well as the other two subgroups automorphic to :

### Direct product and direct factor interpretation

This article describes the subgroup in the group , where is the Klein four-group:

Denoted additively, the four elements of are:

where the addition is modulo 2 in each coordinate separately.

and is the two-element subgroup:

is a normal subgroup (since is an abelian group), so has no other conjugate subgroups. There are two other automorphic subgroups to in :

The quotient group is isomorphic to cyclic group:Z2.

## Cosets

Each of , and is a normal subgroup and each has two cosets: the subgroup itself and the rest of the group. There is a total of six cosets. In fact, *every subset of size two* in the whole group (total of such subsets) is a coset of *exactly* one of these three subgroups.

## Complements

Any two of the three subgroups are permutable complements of each other. Since they're all normal, this means that is an internal direct product of any two of them.

Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|

complemented normal subgroup | normal subgroup with permutable complement | Yes | ||

complemented characteristic subgroup | characteristic subgroup with permutable complement | No | not a characteristic subgroup | |

direct factor | normal subgroup with normal complement | Yes | ||

retract | subgroup with normal complement | Yes | ||

permutably complemented subgroup | subgroup with permutable complement | Yes | ||

lattice-complemented subgroup | subgroup with lattice complement | Yes |

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order of whole group | 4 | |

order of subgroup | 2 | |

index | 2 | |

size of conjugacy class | 1 | |

number of conjugacy classes in automorphism class | 3 | |

size of automorphism class | 3 |

## Effect of subgroup operators

Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|

normalizer | the whole group | -- | Klein four-group |

centralizer | the whole group | -- | Klein four-group |

normal core | the subgroup itself | current page | cyclic group:Z2 |

normal closure | the subgroup itself | current page | cyclic group:Z2 |

characteristic core | trivial subgroup | -- | trivial group |

characteristic closure | the whole group | -- | Klein four-group |

commutator with whole group | trivial subgroup | -- | trivial group |

## Subgroup properties

### Invariance under automorphisms and endomorphisms: properties

Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|

normal subgroup | invariant under all inner automorphisms, equals its conjugate subgroups | Yes | abelian implies every subgroup is normal | |

characteristic subgroup | invariant under all automorphisms | No | Not invariant under automorphism that interchanges coordinates, i.e., that sends to | See also normal not implies characteristic. This is the smallest example of a normal subgroup that is not characteristic and is part of a general family of such examples. |

coprime automorphism-invariant subgroup | invariant under automorphisms of order coprime to that of the group | No | The group has an automorphism of order three that cycles the three non-identity elements -- this cyclically permutes the three automorphs of | |

cofactorial automorphism-invariant subgroup | invariant under all automorphisms whose order has no prime factors other than those of the whole group | No | The coordinate interchange automorphism does not preserve the subgroup, and has order 2 | |

subgroup-cofactorial automorphism-invariant subgroup | invariant under all automorphisms whose order has no prime factors other than those of the subgroup | No | The coordinate interchange automorphism does not preserve the subgroup, and has order 2 | |

retraction-invariant subgroup | invariant under all retractions, i.e., endomorphisms that equal their own square | No | Consider the retraction with kernel and image (realized as projection onto in the realization of as an internal direct product of and ). Under this, maps to . |

### Resemblance-based properties

Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|

order-isomorphic subgroup | isomorphic to any subgroup of the same order | Yes | follows since order is prime, and there is only one isomorphism class of a group of prime order | |

order-automorphic subgroup | any subgroup of the same order is automorphic to it | Yes | the three subgroups of order 2 are precisely | |

isomorph-automorphic subgroup | any isomorphic subgroup is automorphic | Yes | the three subgroups of order 2 are precisely | |

isomorph-conjugate subgroup | conjugate to every isomorphic subgroup | No | distinct normal subgroups -- cannot be conjugate | |

order-conjugate subgroup | conjugate to every subgroup of the same order | No | distinct normal subgroups -- cannot be conjugate | |

automorph-conjugate subgroup | conjugate to every automorphic subgroup | No | distinct normal subgroups -- cannot be conjugate |