# Derived subgroup of nontrivial semidirect product of Z4 and Z4

From Groupprops

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) nontrivial semidirect product of Z4 and Z4 (see subgroup structure of nontrivial semidirect product of Z4 and Z4).

The subgroup is a normal subgroup and the quotient group is isomorphic to direct product of Z4 and 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

Consider the group:

.

This is a group of order 16 with elements:

We are interested in the subgroup:

This is the derived subgroup. In particular, it is a normal subgroup and the quotient group is isomorphic to direct product of Z4 and Z2.

## Cosets

The subgroup is a normal subgroup, so its left cosets coincide with its right cosets. The subgroup has order 2 and index 8, so there are 8 cosets, given as:

## Complements

The subgroup has no permutable complements. Since it is a normal subgroup, this also means it has no lattice complements.

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

complemented normal subgroup | normal subgroup with permutable complement | No | see above | |

permutably complemented subgroup | subgroup with permutable complement | No | ||

lattice-complemented subgroup | subgroup with lattice complement | No | ||

retract | has a normal complement | No | ||

direct factor | normal subgroup with normal complement | No |

## Arithmetic functions

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

order of whole group | 16 | |

order of subgroup | 2 | |

index | 8 | |

size of conjugacy class | 1 | |

number of conjugacy classes in automorphism class | 1 |

## Subgroup-defining functions

Subgroup-defining function | Meaning in general | Why it takes this value |
---|---|---|

derived subgroup | subgroup generated by all commutators | The commutators are precisely the elements of this subgroup. |

## Subgroup properties

### Invariance under automorphisms and endomorphisms

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

central subgroup | contained in the center | Yes | The center is . |

central factor | product with centralizer is whole group | Yes | central implies central factor |

## GAP implementation

The group and subgroup can be constructed as follows, using the SmallGroup and DerivedSubgroup functions:

`G := SmallGroup(16,4); H := DerivedSubgroup(G);`

Implementing this in GAP looks as follows:

gap> G := SmallGroup(16,4); H := DerivedSubgroup(G); <pc group of size 16 with 4 generators> Group([ f3 ])Below is GAP implementation to test some of the assertions in this page:[SHOW MORE]