Abelian implies nilpotent

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., abelian group) must also satisfy the second group property (i.e., nilpotent group)
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Statement

Verbal statement

Any abelian group is a nilpotent group. In fact, abelian groups are precisely the nilpotent groups of nilpotency class equal to 1.

Definitions used

Term	Definition used
abelian group	A group is abelian if it satisfies the following equivalent conditions: its center is the whole group its derived subgroup is trivial.
nilpotent group	A group is nilpotent if it satisfies the following equivalent conditions: The upper central series terminates in finitely many steps, at the whole group. The number of steps is termed the nilpotence class. The lower central series terminates in finitely many steps, at the trivial subgroup. The number of steps is termed the nilpotence class.

Proof

Using upper central series

Given: An Abelian group $G$

To prove: The upper central series of $G$, terminates, after finitely many steps, at $G$

Proof: The first member $Z^{(1)}(G)$ of the upper central series is the center of $G$, which is the whole of $G$ since $G$ is Abelian. Thus the upper central series terminates in 1 step, and $G$ is nilpotent of nilpotence class 1.

Using lower central series

Given: An abelian group $G$

To prove: The lower central series of $G$ terminates in finitely many steps at the trivial subgroup

Proof:The first member of the lower central series of $G$ is the derived subgroup of $G$, which is trivial because $G$ is abelian. Thus the upper central series terminates in 1 step, and $G$ is nilpotent of class 1.

Converse

The converse of this statement is not true: nilpotent not implies abelian.

Intermediate properties

For finite groups

• Dedekind group: This is a group where every subgroup is normal.
• ACIC-group: This is a group where every automorph-conjugate subgroup is characteristic. For finite groups, ACIC implies nilpotent. In general, it does not, but Abelian always implies ACIC.
• Group generated by abelian normal subgroups: This property is weaker than Abelianness. If we require only finitely many Abelian normal subgroups to generate the group, then it is nilpotent. In particular, for finite groups, such a group is always nilpotent.
• Finite group that is 1-isomorphic to an abelian group: This is a finite group for which there exists a bijection to an abelian group (and hence, a finite abelian group) with the property that the bijection is a 1-isomorphism, i.e., it restricts to an isomorphism on cyclic subgroups.
• Finite group that is order statistics-equivalent to an abelian group: This is a finite group whose order statistics match those of a finite baelian group.