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.
Table of contents |
2 Examples 3 Properties |
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. 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}
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 minimial index n for which this happens is the same in both cases.
Examples
As noted above, every abelian group is nilpotent.
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 normal 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 sum of its Sylow subgroupss.