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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Contents
Definition
The group 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:
The subgroup is given as:
Cosets
is a normal subgroup of , so its left cosets coincide with its right cosets. The four cosets are as follows:
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 that plays the role of a dual subgroup to -- in particular, that is isomorphic to the quotient group and its quotient group is isomorphic to . 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 powers where is the underlying prime (here ) | |
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 -- 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 as an extension with abelian normal subgroup and quotient group . Since is abelian, is central, so the action of the quotient group on the normal subgroup is the trivial group action. We can thus study as an extension group arising from a cohomology class for the trivial group action of (which is a Klein four-group) on (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);