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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Statement

We can have a finitely generated abelian group with a subgroup that is characteristic but not fully invariant.

Definitions used

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.

Characteristic subgroup

Further information: Characteristic subgroup

A subgroup $H$ of a group $G$ is termed characteristic in $G$ if, for every automorphism $\sigma$ of $G$, $\sigma (H)=H$.

Fully characteristic subgroup

Further information: Fully invariant subgroup

A subgroup $H$ of a group $G$ is termed fully characteristic in $G$ if, for every endomorphism $\alpha$ of $G$, $\alpha (H)\leq H$.

Related facts

• 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

Proof

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 $G$ be the direct sum of the infinite cyclic group and the cyclic group of order two:

$G:=\mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z}$.

Let $H$ be the cyclic subgroup of $G$ generated by $(2,1)$.

The subgroup is characteristic

Set:

$A:=\{g\in G\mid \exists x\in G,4x=2g\}=\{(2r,s)\mid r\in \mathbb {Z} ,s\in \mathbb {Z} /2\mathbb {Z} \}$

and

$B:=\{g\in G\mid \exists x\in G,g=2x\}=\{(2r,0)\mid r\in \mathbb {Z} \}$.

and:

$C:=\{g\in G\mid \exists x\in G,8x=2g\}=\{(4r,s)\mid r\in \mathbb {Z} ,s\in \mathbb {Z} /2\mathbb {Z} \}$.

Thus, we have:

$D:=A\setminus (B\cup C)=\{(2r,1)\mid r\in \mathbb {Z} ,r\notin 2\mathbb {Z} \}$.

Clearly, any automorphism of $G$ sends $A$ to itself, sends $B$ to itself, and sends $C$ to itself. Thus, any automorphism of $G$ sends $A\setminus (B\cup C)$ to itself. Thus, any automorphism of $G$ sends $D$ to itself. Note that $D$ comprises precisely those elements of $H$ that have the second coordinate equal to $1$: in particular, $(2,1)\in D\subseteq H$, so the subgroup generated by $D$ equals $H$. Thus, any automorphism of $G$ preserves $H$.

The subgroup is not fully invariant

Consider the map:

$\pi _{1}:G\to G,\qquad \pi _{1}(a,b)=(a,0)$.

This map is an endomorphism of $G$, but the image of $(2,1)$ under this map is $(2,0)$, which is not an element of $H$. Thus, $H$ is not fully characteristic in $G$.