Nilpotent group

From Wikipedia, the free encyclopedia

In group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated application of the commutator operation, [x,y] = x^-1y^-1xy. 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 1 Definition 1.1 Explanation of term 2 Examples 3 Properties 4 References

[edit] 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 group of nilpotency class 0, and groups of nilpotency class 1 are exactly non-trivial abelian groups.

[edit] 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 is nilpotent in the sense that the nth iteration of the function is trivial: for all x in G.

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,^[1] 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).

[edit] Examples

• As noted above, every abelian group is nilpotent.
• Finite p-groups are nilpotent (proof).
• 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.
• The Heisenberg group is another example of non-abelian nilpotent group.

[edit] Properties

Since each successive factor group Z_i+1/Z_i 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; in addition, if f is a homomorphism of a nilpotent group of class n, then the image of f is nilpotent of class at most n.

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

• G is a nilpotent group.
• If H is a proper subgroup of G, then H is a proper normal subgroup of N_G(H) (the normalizer of H in G). This is called the normalizer property and can be phrased simply as "normalizers grow".
• Every maximal proper subgroup of G is normal.
• G is the direct product of its Sylow subgroups.

The last statement can be extended to infinite groups: If G is a nilpotent group, then every Sylow subgroup G_p of G is normal, and the direct sum 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.

[edit] References

1. ^ For the term, compare Engel's theorem, also on nilpotency.

This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (January 2008)

• Homology in group theory, by Urs Stammbach, Lecture Notes in Mathematics, Volume 359, Springer-Verlag, New York, 1973, vii+183 pp. review