# Non-central Z4 in M16

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:Z4 and the group is (up to isomorphism) M16 (see subgroup structure of M16).

The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z4.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 consider the group:

with denoting the identity element.

This is a group of order 16, with elements:

We are interested in the subgroup:

This subgroup is isomorphic to cyclic group:Z4, with generator . It is a normal subgroup and the quotient group is also isomorphic to the cyclic group:Z4.

Here is the multiplication table for . Note that is an abelian group so we don't have to worry about left/right issues:

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

## Cosets

The subgroup is a normal subgroup and so its left cosets are the same as its right cosets. It has 4 cosets in the whole group:

Below is the multiplication for . It is a cyclic group of order 4 with generator the coset . Since it is an abelian group, we do not need to worry about left/right convention for the multiplication table:

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

## Subgroup properties

### Invariance under automorphisms and endomorphisms

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

normal subgroup | invariant under inner automorphisms | Yes | Follows from being the only non-central cyclic group of order four. |

characteristic subgroup | invariant under all automorphisms | Yes | Follows from being the only non-central cyclic group of order four. |

fully invariant subgroup | invariant under all endomorphisms | No | The endomorphism (which is a retraction onto ) does not preserve this subgroup. |

isomorph-free subgroup | no other isomorphic subgroup | No | center of M16 is another isomorphic subgroup. |

isomorph-characteristic subgroup | every isomorphic subgroup is characteristic | Yes | The only isomorphic subgroups are the subgroup itself and center of M16. |

## GAP implementation

The group and subgroup pair can be constructed as follows:

`G := SmallGroup(16,6); H := Filtered(NormalSubgroups(G),x -> Order(x) = 4 and IsCyclic(x) and not(x = Center(G)))[1];`

Here is the GAP display:

gap> G := SmallGroup(16,6); H := Filtered(NormalSubgroups(G),x -> Order(x) = 4 and IsCyclic(x) and not(x = Center(G)))[1]; <pc group of size 16 with 4 generators> Group([ f2*f3, f4 ])

Here is GAP code to verify some of the assertions in this page:

gap> Order(G); 16 gap> Order(H); 4 gap> Index(G,H); 4 gap> StructureDescription(H); "C4" gap> StructureDescription(G/H); "C4" gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); false