# First omega subgroup of direct product of Z4 and Z2

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) Klein four-group and the group is (up to isomorphism) direct product of Z4 and Z2 (see subgroup structure of direct product of Z4 and Z2).

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

The group is a direct product of Z4 and Z2, which, for convenience, we describe by ordered pairs with the first members from the integers mod 4 (the first direct factor cyclic group:Z4) and the second member from the integers mod 2 (the second direct factor cyclic group:Z2). It has elements:

The subgroup is given as:

## Cosets

is a subgroup of index two and hence a normal subgroup, so its left cosets and its right cosets coincide. The following are its two cosets:

## Arithmetic functions

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

order of whole group | 8 | |

order of subgroup | 4 | |

index of subgroup | 2 | |

size of conjugacy class of subgroup | 1 | |

number of conjugacy classes in automrophism class of subgroup | 1 | |

size of automorphism class of subgroup | 1 |

## Dual subgroup

We know that subgroup lattice and quotient lattice of finite abelian group are isomorphic, which means that there must exist a subgroup of that plays the role of a *dual* subgroup to -- in particular, that is isomorphic to the quotient group and its quotient group is isomorphic to . The subgroup is first agemo subgroup of direct product of Z4 and Z2.

## Subgroup properties

### Invariance under automorphisms and endomorphisms: basic properties

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

normal subgroup | invariant under all inner automorphisms | Yes | |

characteristic subgroup | invariant under all automorphisms | Yes | |

fully invariant subgroup | invariant under all endomorphisms | Yes |

### Resemblance-based properties and corollaries for invariance under automorphisms and endomorphisms

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

image-closed characteristic subgroup | image under any surjective homomorphism from whole group is characteristic in target group | No | image under quotient map by first agemo subgroup of direct product of Z4 and Z2 gives Z2 in V4. |

verbal subgroup | generated by a set of words | No | Follows from not being image-closed characteristic. |

isomorph-free subgroup | no other isomorphic subgroup | Yes | |

homomorph-containing subgroup | contains every homomorphic image | Yes | |

variety-containing subgroup | contains every subgroup of the whole group in the variety it generates | Yes | In this case, the variety generated by the subgroup is the variety of elementary abelian 2-groups. |

## GAP implementation

The group-subgroup pair can be constructed using the DirectProduct, CyclicGroup, and Filtered functions as follows:

`G := DirectProduct(CyclicGroup(4),CyclicGroup(2)); H := Group(Filtered(G, x -> IsOne(x^2)));`