Levi operator

This article defines a group property modifier (a unary group property operator) -- viz an operator that takes as input a group property and outputs a group property

Definition

Symbol-free definition

The Levi operator is a map from the group property space to the group property space that takes as input a group property $p$ and outputs the property of being a group where every point-closure (viz the normal closure of every element) satisfies the group property $p$ as an abstract group.

Definition with symbols

The Levi operator is a map from the group property space to the group property space that takes as input a group property $p$ and gives as output the property of being a group $G$ such that for any element $x$ in $G$, the normal closure of $x$ satisfies property $p$ as an abstract group.

Application

Important instances of application of the Levi operator:

• Dedekind group: obtained from cyclic group
• 2-Engel group: obtained from Abelian group
• Fitting group: obtained from nilpotent group