Characteristic not implies fully invariant in finitely generated abelian group
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finitely generated Abelian group. That is, it states that in a finitely generated Abelian group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., fully characteristic subgroup)
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Finitely generated Abelian group
Further information: Finitely generated Abelian group
A finitely generated Abelian group is an Abelian group that has a finite generating set. Equivalently, it is a group expressible as a direct sum of finitely many cyclic groups.
Further information: Characteristic subgroup
A subgroup of a group is termed characteristic in if, for every automorphism of , .
Fully characteristic subgroup
Further information: Fully invariant subgroup
A subgroup of a group is termed fully characteristic in if, for every endomorphism of , .
- Characteristic equals fully invariant in odd-order abelian group
- Characteristic not implies fully invariant in finite abelian group
- Characteristic not implies fully invariant
- Characteristic direct factor not implies fully invariant
- Characteristic equals verbal in free abelian group
There is essentially only one kind of counterexample. We describe this counterexample for the infinite case. A finite version of this counterexample can be found at characteristic not implies fully invariant in finite abelian group.
Let be the direct sum of the infinite cyclic group and the cyclic group of order two:
Let be the cyclic subgroup of generated by .
The subgroup is characteristic
Thus, we have:
Clearly, any automorphism of sends to itself, sends to itself, and sends to itself. Thus, any automorphism of sends to itself. Thus, any automorphism of sends to itself. Note that comprises precisely those elements of that have the second coordinate equal to : in particular, , so the subgroup generated by equals . Thus, any automorphism of preserves .
The subgroup is not fully invariant
Consider the map:
This map is an endomorphism of , but the image of under this map is , which is not an element of . Thus, is not fully characteristic in .