# Nilpotent group

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*^{-1}*y*^{-1}*xy*. Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups.

## Definition

We start by defining the *lower central series* of a group *G* as a series of groups *G* = *A*_{0}, *A*_{1}, *A*_{2}, ..., *A*_{i}, ..., where each *A*_{i+1} = [*A*_{i}, *G*], the subgroup of *G* generated by all commutators [*x*,*y*] with *x* in *A*_{i} and *y* in *G*. Thus, *A*_{1} = [*G*,*G*] = *G*^{1}, the commutator subgroup of *G*; *A*_{2} = [*G*^{1}, *G*], etc.

If *G* is abelian, then [*G*,*G*] = *E*, the trivial subgroup. As an extension of this idea, we call a group *G* **nilpotent** if there is some natural number *n* such that *A*_{n} is trivial. If *n* is the smallest natural number such that *A*_{n} is trivial, then we say that *G* is *nilpotent of class* *n*. Every abelian group is nilpotent of class 1, except for the trivial group, which is nilpotent of class 0. If a group is nilpotent of class at most *m*, then it is sometimes called a nil-*m* group.

For a justification of the term *nilpotent*, start with a nilpotent group *G*, an element *g* of *G* and define a function *f* : *G* → *G* by *f*(*x*) = [*x*,*g*]. Then this function is nilpotent in the sense that there exists a natural number *n* such that *f*^{n}, the *n*-th iteration of *f*, sends every element *x* of *G* to the identity element.

An equivalent definition of a nilpotent group is arrived at by way of the *upper central series* of *G*, which is a sequence of groups *E* = *Z*_{0}, *Z*_{1}, *Z*_{2}, ..., *Z*_{i}, ..., where each successive group is defined by:

*Z*_{i+1}= {*x*in*G*: [*x*,*y*] in*Z*_{i}for all*y*in*G*}

In this case, *Z*_{1} is the center of *G*, and for each successive group, the factor group *Z*_{i+1}/*Z*_{i} is the center of *G*/*Z*_{i}. For an abelian group, *Z*_{1} is simply *G*; a group is called nilpotent of class *n* if *Z*_{n} = *G* for a minimal *n*.

These two definitions are equivalent: the lower central series reaches the trivial subgroup *E* if and only if the upper central series reaches *G*; furthermore, the minimal index *n* for which this happens is the same in both cases.

## Examples

As noted above, every abelian group is nilpotent.

For a small non-abelian example, consider the quaternion group *Q*_{8}. It has center {1, −1} of order 2, and its lower central series is {1}, {1, −1}, *Q*_{8}; so it is nilpotent of class 2. In fact, every direct sum of finite *p*-groups is nilpotent.

The discrete Heisenberg group is another example of non-abelian nilpotent group.

## 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*(*H*) (the normalizer of*H*in*G*). - 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).