First omega 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) Klein four-group 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:Z2.
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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 subgroup of index two and hence a normal subgroup, so its left cosets and its right cosets coincide. The following are its two cosets:
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 8 | |
order of subgroup | 4 | |
index of subgroup | 2 | |
size of conjugacy class of subgroup | 1 | |
number of conjugacy classes in automrophism 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 agemo subgroup of direct product of Z4 and Z2.
Subgroup properties
Invariance under automorphisms and endomorphisms: basic properties
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
normal subgroup | invariant under all inner automorphisms | Yes | |
characteristic subgroup | invariant under all automorphisms | Yes | |
fully invariant subgroup | invariant under all endomorphisms | Yes |
Resemblance-based properties and corollaries for invariance under automorphisms and endomorphisms
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
image-closed characteristic subgroup | image under any surjective homomorphism from whole group is characteristic in target group | No | image under quotient map by first agemo subgroup of direct product of Z4 and Z2 gives Z2 in V4. |
verbal subgroup | generated by a set of words | No | Follows from not being image-closed characteristic. |
isomorph-free subgroup | no other isomorphic subgroup | Yes | |
homomorph-containing subgroup | contains every homomorphic image | Yes | |
variety-containing subgroup | contains every subgroup of the whole group in the variety it generates | Yes | In this case, the variety generated by the subgroup is the variety of elementary abelian 2-groups. |
GAP implementation
The group-subgroup pair can be constructed using the DirectProduct, CyclicGroup, and Filtered functions as follows:
G := DirectProduct(CyclicGroup(4),CyclicGroup(2)); H := Group(Filtered(G, x -> IsOne(x^2)));