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:
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

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.

Converse

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

Intermediate properties

For finite groups