Nilpotent Groups

Introduction

Nilpotent groups are a special subclass of solvable groups that are "almost abelian." They have a more constrained structure than solvable groups, making them closer to abelian groups.

Definition

Definition 8.4: A group $G$ is nilpotent if its upper central series terminates at $G$ . The upper central series is defined recursively: $Z_{0} (G) = {e}$ , and $Z_{i + 1} (G)$ is the subgroup of $G$ such that $Z_{i + 1} (G) / Z_{i} (G) = Z (G / Z_{i} (G))$ .

Alternatively, a group is nilpotent if its lower central series terminates at ${e}$ , where $γ_{1} (G) = G$ and $γ_{i + 1} (G) = [G, γ_{i} (G)]$ .

Properties

All nilpotent groups are solvable, but the converse is not true ( $S_{3}$ is solvable but not nilpotent)
All finite $p$ -groups are nilpotent
A key property of finite nilpotent groups is that they are the direct product of their Sylow subgroups

Nilpotent groups have a more constrained structure than solvable groups, making them closer to abelian groups. For example, in a finite nilpotent group, any two elements with relatively prime orders must commute.

Examples

Example 1: Abelian Groups

All abelian groups are both solvable and nilpotent. For an abelian group $G$ , we have $G^{'} = {e}$ , so the derived series terminates immediately.

Example 2: p-Groups

All finite $p$ -groups are nilpotent. For example, the quaternion group $Q_{8}$ is nilpotent:

$Z_{0} (Q_{8}) = {e}$
$Z_{1} (Q_{8}) = ⟨ - 1 ⟩$
$Z_{2} (Q_{8}) = Q_{8}$

Example 3: Direct Products

The direct product of nilpotent groups is nilpotent.

Example 4: Non-Nilpotent Solvable Groups

The symmetric group $S_{3}$ is solvable but not nilpotent:

$S_{3}^{'} = A_{3}$
$S_{3}^{″} = {e}$
But the upper central series never reaches $S_{3}$

Applications

Application 1: Group Classification

Nilpotent groups are important in the classification of finite groups. Many important families of groups (such as $p$ -groups) are nilpotent, and understanding their structure is crucial for the classification of all finite groups.

Application 2: Representation Theory

Nilpotent groups have particularly nice properties in representation theory. For example, all irreducible representations of a finite nilpotent group are monomial (induced from one-dimensional representations of subgroups).

Application 3: Structure Analysis

The constrained structure of nilpotent groups makes them easier to analyze than general solvable groups.

Supersolvable Groups

A group $G$ is supersolvable if it has a normal series with cyclic factors. Supersolvable groups are a subclass of solvable groups that are even more constrained than nilpotent groups.