Diagonally embedded Z4 in direct product of Z8 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:Z4 and the group is (up to isomorphism) direct product of Z8 and Z2 (see subgroup structure of direct product of Z8 and Z2).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z4.
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Here, the group is the group direct product of Z8 and Z2, written for convenience using ordered pairs with the first element an integer mod 8 (coming from cyclic group:Z8) and the second element an integer mod 2. The addition is coordinate-wise.
has 16 elements:
The subgroup is:
Invariance under automorphisms and endomorphisms: basic properties
|normal subgroup||invariant under inner automorphisms||Yes||abelian implies every subgroup is normal|
|characteristic subgroup||invariant under automorphisms||Yes||The elements and are the only elements of the group with the property that the square of the element has order two but the element is not itself a square. Any automorphism must therefore preserve or interchange these two elements, and hence must send (which is cyclic on both of them) to itself.|
|fully invariant subgroup||invariant under all endomorphisms||No|| The projection to the second direct factor (map ) does not preserve this subgroup. Nor does the projection to the first direct factor.|
This is the smallest example for characteristic not implies fully invariant in finite abelian group. There are no analogous examples for odd order, because characteristic equals fully invariant in odd-order abelian group.
Invariance under automorphisms and endomorphisms: advanced properties
|square-closed characteristic subgroup||The square of (direct product ) is characteristic in .||No||PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]|
|finite direct power-closed characteristic subgroup||Any finite direct power of is characteristic in the corresponding finite direct power of .||No||Follows from not being square-closed characteristic.|