# Central subgroup generated by a non-square in 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 quaternion group.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:

The quotient group is isomorphic to quaternion group.

## 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:

## 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 properties

### Invariance under automorphisms and endomorphisms

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

normal subgroup | invariant under inner automorphisms | Yes | central implies normal |

characteristic subgroup | invariant under all automorphisms | Yes | The non-identity element is the unique element in the center that is not a square. |

fully invariant subgroup | invariant under all endomorphisms | No | The endomorphism sends this subgroup to , the derived subgroup of nontrivial semidirect product of Z4 and Z4. |

## GAP implementation

The group and subgroup pair can be constructed using GAP as follows:

`G := SmallGroup(16,4); H := Group(Difference(Set(Center(G)),Set(List(G,x -> x^2))));`

The GAP display looks as below:

gap> G := SmallGroup(16,4); H := Group(Difference(Set(Center(G)),Set(List(G,x -> x^2)))); <pc group of size 16 with 4 generators> <pc group with 1 generators>

Here is some GAP code verifying some of the assertions on this page:

gap> Order(G); 16 gap> Order(H); 2 gap> Index(G,H); 8 gap> StructureDescription(H); "C2" gap> StructureDescription(G/H); "Q8" gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); false