Nilpotent Group - Definition

Definition

The definition uses the idea, explained on its own page, of a central series for a group. The following are equivalent formulations:

• A nilpotent group is one that has a central series of finite length.
• A nilpotent group is one whose lower central series terminates in the trivial subgroup after finitely many steps.
• A nilpotent group is one whose upper central series terminates in the whole group after finitely many steps.

For a nilpotent group, the smallest n such that G has a central series of length n is called the nilpotency class of G ; and G is said to be nilpotent of class n. (By definition, the length is n if there are n + 1 different subgroups in the series, including the trivial subgroup and the whole group.)

Equivalently, the nilpotency class of G equals the length of the lower central series or upper central series. If a group has nilpotency class at most m, then it is sometimes called a nil-m group.

It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class 0, and groups of nilpotency class 1 are exactly the non-trivial abelian groups.