First agemo subgroup of direct product of Z4 and Z2

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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);