nLab nilpotent group

Idea

A group is nilpotent if it can be built up by central extensions from abelian groups. A central series for a group is a witness to its nilpotency.

More generally, we may speak about $\mathcal{A}$-nilpotent $\pi$-groups, for any class $\mathcal{A}$ of abelian groups and any group $\pi$ acting on our groups. A group $G$ is equivalently an Ab-nilpotent $G$-group for its adjoint action.

Definition

Nilpotent groups and central series

Definition

The class of nilpotent groups is defined inductively by the following clauses:

1. The trivial group $1$ is nilpotent.

2. If $1\to G' \to G \to G''\to 1$ is a central extension (so that in particular, $G'$ is abelian) and $G''$ is nilpotent, then $G$ is nilpotent.

Phrased in this way, nilpotency is an inductive predicate on the class of groups. If we regard the same clauses as defining an inductive family indexed over the class of groups, then we obtain the definition of a central series.

Definition

The set of central series for a group $G$ is defined inductively by the following clauses:

1. The trivial group $1$ has a specified central series, called the “trivial” one.

2. From any central extension $1\to G' \to G \to G''\to 1$ and any central series of $G''$, we obtain a central series of $G$, called an “extension” of the given one.

Thus, central series are “witnesses” to nilpotency: a group is nilpotent if and only if it has some central series.

If we “expand out” the inductive definition of central series, and use the isomorphism theorems?, we see that it consists of a sequence of central extensions

$$1 \to G_1 \to G \to G/G_1 \to 1$$

$$1 \to G_2/G_1 \to G/G_1 \to G/G_2 \to 1$$

$$1 \to G_3/G_2 \to G/G_2 \to G/G_3 \to 1$$

$$\dots$$

$$1 \to G/G_{n-1} \to G/G_{n-1} \to 1 \to 1$$

and therefore a sequence of normal subgroups

$$1 = G_0 \trianglelefteq G_1 \trianglelefteq G_2 \trianglelefteq \dots \trianglelefteq G_{n-1} \trianglelefteq G_n = G$$

such that each $G_i/G_{i-1}$ is central in $G/G_{i-1}$. This is the “usual” definition of central series.

Nilpotency class

Every central series has a length, defined recursively by saying that the length of the trivial central series is $0$ and the length of an extension is one more than the length of the original. The nilpotency class of a nilpotent group is the minimum length of all of its central series.

Proofs about nilpotent groups are often most naturally phrased using induction over the inductive definition of nilpotency. However, probably due to widespread ignorance about inductive definitions, it is common to find them phrased instead using ordinary natural-number induction over the nilpotency class.

Co-nilpotent groups and central streams

If we interpret the same defining clauses of nilpotent groups and central series coinductively rather than inductively, we obtain notions that might be called co-nilpotent groups and central streams (“stream” being the standard name for the coinductive counterpart of a list). Explicitly, a central stream is a descending countable sequence of normal subgroups, such that each successive quotient is central in the corresponding quotient of the whole group, that may or may not ever terminate with the trivial group.

In fact, every group admits some central stream and hence is co-nilpotent. Two canonical central streams associated to any group are its lower central series and its upper central series (for now see Wikipedia). Despite the names, these two central streams are actually central series (i.e. they terminate at the trivial group) if and only if the group is nilpotent.

$\mathcal{A}$-nilpotent $\pi$-groups

The following generalization of nilpotent groups is sometimes useful. Let $\mathcal{A}$ be any class of abelian groups containing $0$, and let $\pi$ be any group. By a $\pi$-group we mean a group with an action of $\pi$ (through group automorphisms, of course).

Definition

The class of $\mathcal{A}$-nilpotent $\pi$-groups is defined inductively by the following clauses:

1. The trivial group $1$ is $\mathcal{A}$-nilpotent.

2. If $1\to G' \to G \to G''\to 1$ is a central extension of $\pi$-groups (i.e. a central extension whose maps are $\pi$-equivariant), and $G''$ is $\mathcal{A}$-nilpotent while $G'\in \mathcal{A}$ and $\pi$ acts trivially on $G'$, then $G$ is $\mathcal{A}$-nilpotent.

A group $G$ is nilpotent in the original sense if and only if it is an $Ab$-nilpotent $G$-group with its adjoint action. If $\pi$ is nontrivial and/or $\mathcal{A}$ is strictly smaller than $Ab$, then the notion of $\mathcal{A}$-nilpotency can be nontrivial even for abelian groups (whereas every abelian group is obviously nilpotent in the ordinary sense).

Properties

Every nilpotent group is an example of a solvable group (indeed, the groups in the lower central series of any group can be term-wise included into its derived series).

The Sylow p-subgroups of any nilpotent group are normal. The direct product of these subgroups in such a group is its torsion subgroup.

Examples

Generally:

• Every abelian group is nilpotent.

• The direct product group of two nilpotent groups is again nilpotent.

• The central extension of any abelian group is nilpotent, a famous class of examples of such are the Heisenberg groups.

Specifically:

• The multiplicative group of upper triangular unipotent $n \times n$ matrices with coefficients in any field is nilpotent.

Related pages

• solvable group

• nilpotent module

• nilpotent element

• nilpotent Lie algebra

• nilpotent topological space

• localization of a space

References

• Peter Hilton, Nilpotency in group theory and topology, Publicacions de la Secció de Matemàtiques Vol. 26, No. 3 (1982), pp. 47-78 (jstor:43741908)

• Peter May, Kate Ponto, §3.1 in: More concise algebraic topology, University of Chicago Press (2012) [ISBN:9780226511795, pdf]

See also:

• eom nilpotent group

• Wikipedia nilpotent group

On nilpotent Lie groups (cf. at nilpotent Lie algebra):

• Arthur A. Sagle, Ralph E. Walde: Nilpotent Lie Groups and Algebras, Chapter 11 in: Introduction to Lie Groups and Lie Algebras, Pure and Applied Mathematics 51, Elsevier (1973) 215-227 [doi:10.1016/S0079-8169(08)61671-2]

• Laurence Corwin, Frederick P. Greenleaf: Representations of Nilpotent Lie Groups and their Applications – Volume 1 Part 1. Basic Theory and Examples, Cambridge University Press (2004) [ISBN:9780521604956]

• Veronique Fischer, Michael Ruzhansky: Quantization on Nilpotent Lie Groups, Birkhäuser (2016) [doi:10.1007/978-3-319-29558-9]

• Manuel Amann, On quasi-isometric nilpotent Lie groups [arXiv:1710.04542]