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.
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

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));