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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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:
|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|
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.
Invariance under automorphisms and endomorphisms: basic properties
|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
|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.|
G := DirectProduct(CyclicGroup(4),CyclicGroup(2)); H := Group(Filtered(G, x -> IsOne(x^2)));