## Nilpotent groups

Abbreviation: **NGrp**

### Definition

A ** nilpotent group** is a group $\mathbf{G}=\langle G,\cdot,^{-1},1\rangle$ that is

** nilpotent**: if $Z_0=\{1\}$ and $\forall i(Z_{i+1}=\{x \in G : \forall y\ xyx^{-1}y^{-1} \in Z_i\})$ then $\exists n(Z_n=G)$

Remark: Note that $Z_1=Z(G)$, the center of $G$. The smallest $n$ for which $Z_n=G$ is the ** nilpotence class of $G$**. E.g. Abelian
groups are of nilpotence class 1.

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be nilpotent groups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$

### Definition

An ** …** is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$

### Examples

Example 1:

### Basic results

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

### Subclasses

[[Abelian groups]]

### Superclasses

[[Solvable groups]] supervariety

### References

Trace: » nilpotent_groups