Nilpotent group

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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.

[edit] Definition

We start by defining the lower (or descending) 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, 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 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 (or ascending) central series of G, which is a sequence of groups E = Z₀, Z₁, Z₂, ..., Z_i, ..., where each successive group is defined by:

In this case, Z₁ is the centre of G, and for each successive group, the factor group Z_i+1/Z_i is the centre 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 minimal index n for which this happens is the same in both cases.

[edit] 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 product of finitely many finite p-groups is nilpotent.

The discrete 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(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).