# Characteristic not implies injective endomorphism-invariant in finitely generated abelian group

From Groupprops

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) neednotsatisfy the second subgroup property (i.e., injective endomorphism-invariant subgroup)

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## Statement

It is possible to have a finitely generated abelian group and a subgroup of such that is a characteristic subgroup of but is *not* an injective endomorphism-invariant subgroup of , i.e., there exists an injective endomorphism of such that is not contained in .

## Related facts

- Characteristic not implies injective endomorphism-invariant
- Center not is injective endomorphism-invariant
- Characteristic not implies fully invariant in finite abelian group

## Proof

`Further information: direct product of Z and Z2`

Suppose is the direct product of the group of integers and the cyclic group . Suppose is the subgroup generated by the element .

- is a characteristic subgroup of : Let , , , and . Then, the set is the two-element set . Both of these are generators of the subgroup . Hence, is a characteristic subgroup of .
- is not an injective endomorphism-invariant subgroup of : Consider the injective endomorphism defined by . Then, is not contained in , because , which is not in .