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^-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.

Definition

We start by defining the lower central series of a group G as a series of groups G = A₀, A₁, A₂, ..., 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, A₁ = [G,G] = G¹, the commutator subgroup of G; A₂ = [G¹, 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 fⁿ, 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 = Z₀, Z₁, Z₂, ..., 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, Z₁ 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, Z₁ 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 minimial 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 Q₈. It has center {1, -1} of order 2, and its lower central series is {1}, {1, -1}, Q₈; so it is nilpotent of class 2. In fact, every direct sum of finite p-groups 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 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).