First agemo 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) 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 Klein four-group.
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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:

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

Cosets

$H$ is a normal subgroup of $G$, so its left cosets coincide with its right cosets. The four cosets are as follows:

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

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	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 omega subgroup of direct product of Z4 and Z2.

Subgroup-defining function

Subgroup-defining function	Meaning in general	Why it takes this value
first agemo subgroup	subgroup generated by all $p^{th}$ powers where $p$ is the underlying prime (here $p = 2$)
Frattini subgroup	intersection of all maximal subgroups
Jacobson radical	intersection of all maximal normal subgroups	Jacobson radical coincide with Frattini subgroup for a group of prime power order and more generally a nilpotent group because the maximal subgroups are precisely the maximal normal subgroups.

Subgroup properties

Invariance under automorphisms and endomorphisms: basic properties

Property	Meaning	Satisfied?	Explanation	GAP verification (set G := DirectProduct(CyclicGroup(4),CyclicGroup(2)); H := Agemo(G,2,1);)
normal subgroup	invariant under all inner automorphisms	Yes		IsNormal(G,H); using IsNormal
characteristic subgroup	invariant under all automorphisms	Yes		IsCharacteristicSubgroup(G,H); using IsCharacteristicSubgroup
fully invariant subgroup	invariant under all endomorphisms	Yes		IsFullinvariant(G,H); using IsFullinvariant

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

Property	Meaning	Satisfied?	Explanation
verbal subgroup	generated by a set of words	Yes	precisely the set of squares
isomorph-free subgroup	no other isomorphic subgroup	No	there are other isomorphic subgroups such as $\{ (0,0), (0,1) \}$ -- see non-characteristic order two subgroups of direct product of Z4 and Z2
homomorph-containing subgroup	contains every homomorphic image	No	Follows from not being isomorph-free
intermediately characteristic subgroup	characteristic in every intermediate subgroup	No	not characteristic inside first omega subgroup of direct product of Z4 and Z2, in there, it looks like Z2 in V4

Cohomology interpretation

We can think of $G$ as an extension with abelian normal subgroup $H$ and quotient group $G/H$. Since $G$ is abelian, $H$ is central, so the action of the quotient group on the normal subgroup is the trivial group action. We can thus study $G$ as an extension group arising from a cohomology class for the trivial group action of $G/H$ (which is a Klein four-group) on $H$ (which is cyclic group:Z2).

For more, see second cohomology group for trivial group action of V4 on Z2.

GAP implementation

The group-subgroup pair can be defined using the DirectProduct, CyclicGroup, and Agemo functions:

G := DirectProduct(CyclicGroup(4),CyclicGroup(2)); H := Agemo(G,2,1);