Characteristic not implies fully invariant in finite 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 finite abelian group. That is, it states that in a finite abelian group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., fully invariant subgroup)
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- Characteristic equals fully invariant in odd-order abelian group
- Characteristic not implies fully invariant in finitely generated abelian group
- Characteristic equals verbal in free 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 invariant in .