nLab
nilpotent group

Nilpotent groups

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 π’œ\mathcal{A}-nilpotent Ο€\pi-groups, for any class π’œ\mathcal{A} of abelian groups and any group Ο€\pi acting on our groups. A group GG is equivalently an Ab-nilpotent GG-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 11 is nilpotent.

  2. If 1→G′→G→G″→11\to G' \to G \to G''\to 1 is a central extension (so that in particular, G′G' is abelian) and G″G'' is nilpotent, then GG 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 GG is defined inductively by the following clauses:

  1. The trivial group 11 has a specified central series, called the β€œtrivial” one.

  2. From any central extension 1β†’Gβ€²β†’Gβ†’Gβ€³β†’11\to G' \to G \to G''\to 1 and any central series of Gβ€³G'', we obtain a central series of GG, called an β€œextension” of the given one.

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

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

1→G 1→G→G/G 1→1 1 \to G_1 \to G \to G/G_1 \to 1
1β†’G 2/G 1β†’G/G 1β†’G/G 2β†’1 1 \to G_2/G_1 \to G/G_1 \to G/G_2 \to 1
1β†’G 3/G 2β†’G/G 2β†’G/G 2β†’1 1 \to G_3/G_2 \to G/G_2 \to G/G_2 \to 1
… \dots
1β†’G/G nβˆ’1β†’G/G nβˆ’1β†’1β†’1 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⊴G 1⊴G 2βŠ΄β€¦βŠ΄G nβˆ’1⊴G 0=G 1 = G_0 \trianglelefteq G_1 \trianglelefteq G_2 \trianglelefteq \dots \trianglelefteq G_{n-1} \trianglelefteq G_0 = G

such that each G i/G iβˆ’1G_i/G_{i-1} is central in G/G iβˆ’1G/G_{i-1}. This is the β€œusual” 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 00 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 (β€œstream” 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.

π’œ\mathcal{A}-nilpotent Ο€\pi-groups

The following generalization of nilpotent groups is sometimes useful. Let π’œ\mathcal{A} be any class of abelian groups containing 00, and let Ο€\pi be any group. By a Ο€\pi-group we mean a group with an action of Ο€\pi (through group automorphisms, of course).

Definition

The class of π’œ\mathcal{A}-nilpotent Ο€\pi-groups is defined inductively by the following clauses:

  1. The trivial group 11 is π’œ\mathcal{A}-nilpotent.

  2. If 1β†’Gβ€²β†’Gβ†’Gβ€³β†’11\to G' \to G \to G''\to 1 is a central extension of Ο€\pi-groups (i.e. a central extension whose maps are Ο€\pi-equivariant), and Gβ€³G'' is π’œ\mathcal{A}-nilpotent while Gβ€²βˆˆπ’œG'\in \mathcal{A} and Ο€\pi acts trivially on Gβ€²G', then GG is π’œ\mathcal{A}-nilpotent.

A group GG is nilpotent in the original sense if and only if it is an AbAb-nilpotent GG-group with its adjoint action. If Ο€\pi is nontrivial and/or π’œ\mathcal{A} is strictly smaller than AbAb, then the notion of π’œ\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).

References

Revised on June 26, 2014 17:49:44 by Toby Bartels (75.88.85.62)