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.

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 …: $axiom$

Example 1:

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.

Classtype higher-order yes yes, $n=2$ yes yes

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