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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|abelian group||A group is abelian if it satisfies the following equivalent conditions:|
|nilpotent group||A group is nilpotent if it satisfies the following equivalent conditions:|
Using upper central series
Given: An Abelian group
To prove: The upper central series of , terminates, after finitely many steps, at
Proof: The first member of the upper central series is the center of , which is the whole of since is Abelian. Thus the upper central series terminates in 1 step, and is nilpotent of nilpotence class 1.
Using lower central series
Given: An abelian group
To prove: The lower central series of terminates in finitely many steps at the trivial subgroup
Proof:The first member of the lower central series of is the derived subgroup of , which is trivial because is abelian. Thus the upper central series terminates in 1 step, and is nilpotent of class 1.
The converse of this statement is not true: nilpotent not implies abelian.
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.