Nilpotent group

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In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable.

Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups.

Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series.

Contents

Definition

The following are equivalent definitions for a nilpotent group:

  • 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. Equivalently, the nilpotency class of G equals the length of the lower central series or upper central series (the minimum n such that the nth term is the trivial subgroup, respectively whole group). If a group has nilpotency class at most m, then it is sometimes called a nil-m group.

The trivial group is the unique[citation needed] group of nilpotency class 0, and groups of nilpotency class 1 are exactly non-trivial abelian groups.[1][2]


Examples

  • As noted above, every abelian group is nilpotent.[1][3]
  • For a small non-abelian example, consider the quaternion group Q8, which is a smallest non-abelian p-group. It has center {1, −1} of order 2, and its upper central series is {1}, {1, −1}, Q8; so it is nilpotent of class 2.
  • All finite p-groups are in fact nilpotent (proof). The maximal class of a group of order pn is n-1. The 2-groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups.
  • The direct product of two nilpotent groups is nilpotent.[4]
  • Conversely, every finite nilpotent group is the direct product of p-groups.[5]
  • The Heisenberg group is an example of non-abelian[6], infinite nilpotent group[7].

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