# Non-characteristic order two subgroups 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) cyclic group:Z2 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:Z4.
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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: $\! \{ (0,0), (1,0), (2,0), (3,0), (0,1), (1,1), (2,1), (3,1) \}$

We are interested in two subgroups $H_1$ and $H_2$, both isomorphic to cyclic group:Z2, that are automorphic subgroups (i.e., $H_1$ can be sent to $H_2$ by an automorphism of $G$). We have: $H_1 = \{ (0,0), (0,1) \}$ $H_2 = \{ (0,0), (2,1) \}$

The corresponding quotient groups in both cases are isomorphic to cyclic group:Z4.

Note that both of these are direct factors of the whole group. They are precisely the non-characteristic subgroups of the whole group of order two. There is also a characteristic subgroup of order two, given by $\{ (0,0), (2,0) \}$, which is described on the page first agemo subgroup of direct product of Z4 and Z2.

## Cosets $H_1$ is a normal subgroup of $G$, so its left cosets coincide with its right cosets. The four cosets are as follows: $\! \{ (0,0), (0,1) \}, \{ (1,0), (1,1) \}, \{ (2,0), (2,1) \}, \{ (3,0), (3,1) \}$ $H_2$ is a normal subgroup of $G$, so its left cosets coincide with its right cosets. The four cosets are as follows: $\! \{ (0,0), (2,1) \}, \{ (1,0), (3,1) \}, \{ (2,0), (0,1) \}, \{ (3,0), (1,1) \}$

## Arithmetic functions

Function Value Explanation
order of whole group 8
order of subgroup 2
index of subgroup 4
size of conjugacy class of subgroup 1
number of conjugacy classes in automorphism class of subgroup 2
size of automorphism class of subgroup 2

## Dual subgroup

We know that subgroup lattice and quotient lattice of finite abelian group are isomorphic, which means that there must exist an automorphism class of subgroups of $G$ that plays the role of dual subgroups to $H_1$ and $H_2$ -- in particular, that is isomorphic to the quotient group $G/H_1$ and its quotient group is isomorphic to $H_1$. This automorphism class of subgroups is given by Z4 in direct product of Z4 and Z2.

## Subgroup properties

### Invariance under automorphisms and endomorphisms: basic properties

Property Meaning Satisfied? Explanation GAP verification (set G := DirectProduct(CyclicGroup(4),CyclicGroup(2)); H := Image(Embedding(G,2));)
normal subgroup invariant under all inner automorphisms Yes IsNormal(G,H); using IsNormal
characteristic subgroup invariant under all automorphisms No IsCharacteristicSubgroup(G,H); using IsCharacteristicSubgroup
fully invariant subgroup invariant under all endomorphisms No IsFullinvariant(G,H); using IsFullinvariant

### Factorization-related properties

Property Meaning Satisfied? Explanation
direct factor one of the factors in an internal direct product Yes The complement is $\{ (0,0), (1,0), (2,0), (3,0) \}$
retract has a normal complement Yes Follows from being a direct factor
complemented normal subgroup has a normal complement Yes Follows from being a direct factor
permutably complemented subgroup has a normal complement Yes Follows from being a direct factor
AEP-subgroup any automorphism of the subgroup extends to an automorphism of the whole group Yes direct factor implies AEP

## GAP implementation

The group-subgroup pair can be defined using the DirectProduct, CyclicGroup, Image, and Embedding as:

G := DirectProduct(CyclicGroup(4),CyclicGroup(2)); H := Image(Embedding(G,2));