# 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.
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## Definition

The group $G$ 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:

$\! \{ (0,0), (1,0), (2,0), (3,0), (0,1), (1,1), (2,1), (3,1) \}$

The subgroup $H$ is given as:

$\! \{ (0,0), (2,0), (0,1), (2,1) \}$

## Cosets

$H$ 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:

$\! H = \{ (0,0), (2,0), (0,1), (2,1) \}, G \setminus H = \{ (1,0), (1,1), (3,0), (3,1) \}$

## 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 $G$ that plays the role of a dual subgroup to $H$ -- in particular, that is isomorphic to the quotient group $G/H$ and its quotient group is isomorphic to $H$. 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)));