Finite characteristic series 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

Definition with symbols

The finite characteristic series operator is a group property modifier that takes as input a group property $p$ and outputs a group property $q$ such that a group $G$ satisfies $q$ if and only if it has a characteristic series $e=H_0\le H_1\le \dots le{H}_r=G$ such that each quotient $H_i/H_{i-1}$ satisfies property $p$.

Relation with other modifiers

Weaker modifiers

• Finite normal series operator
• Poly operator

Properties

This group property modifier is monotone, viz if $p\le q$ are group properties and $f$ is the operator, then $f(p)\le f(q)$

This group property modifier is ascendant, viz the image of any group property under this modifier is always weaker than the group property we started with