Nilpotent group

In 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. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.[1]

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.


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

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

Equivalently, the nilpotency class of equals the length of the lower central series or upper central series. If a group has nilpotency class at most , then it is sometimes called a nil- 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.[2][3]


A portion of the Cayley graph of the discrete Heisenberg group, a well-known nilpotent group.

Explanation of term

Nilpotent groups are so called because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group G of nilpotence degree n and an element g, the function defined by (where is the commutator of g and x) is nilpotent in the sense that the nth iteration of the function is trivial: for all in .

This is not a defining characteristic of nilpotent groups: groups for which is nilpotent of degree n (in the sense above) are called n-Engel groups,[9] and need not be nilpotent in general. They are proven to be nilpotent if they have finite order, and are conjectured to be nilpotent as long as they are finitely generated.

An abelian group is precisely one for which the adjoint action is not just nilpotent but trivial (a 1-Engel group).


Since each successive factor group Zi+1/Zi in the upper central series is abelian, and the series is finite, every nilpotent group is a solvable group with a relatively simple structure.

Every subgroup of a nilpotent group of class n is nilpotent of class at most n;[10] in addition, if f is a homomorphism of a nilpotent group of class n, then the image of f is nilpotent[10] of class at most n.

The following statements are equivalent for finite groups,[11] revealing some useful properties of nilpotency:

The last statement can be extended to infinite groups: if G is a nilpotent group, then every Sylow subgroup Gp of G is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in G (see torsion subgroup).

Many properties of nilpotent groups are shared by hypercentral groups.


  1. Dixon, M. R.; Kirichenko, V. V.; Kurdachenko, L. A.; Otal, J.; Semko, N. N.; Shemetkov, L. A.; Subbotin, I. Ya. (2012). "S. N. Chernikov and the development of infinite group theory". Algebra and Discrete Mathematics. 13 (2): 169-208.
  2. 1 2 Suprunenko (1976). Matrix Groups. p. 205.
  3. Tabachnikova & Smith (2000). Topics in Group Theory (Springer Undergraduate Mathematics Series). p. 169.
  4. Hungerford (1974). Algebra. p. 100.
  5. Zassenhaus (1999). The theory of groups. p. 143.
  6. Zassenhaus (1999). Theorem 11. p. 143.
  7. Haeseler (2002). Automatic Sequences (De Gruyter Expositions in Mathematics, 36). p. 15.
  8. Palmer (2001). Banach algebras and the general theory of *-algebras. p. 1283.
  9. For the term, compare Engel's theorem, also on nilpotency.
  10. 1 2 Bechtell (1971), p. 51, Theorem 5.1.3
  11. Isaacs (2008), Thm. 1.26


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