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
The poly operator takes as input a group property and outputs the Kleene star closure of with respect to the extension operator, bracketed on the left.
Definition with symbols
Given a group property , the poly operator gives the group property defined as follows:
A group has property if we can find a subnormal series such that each satisfies property as an abstract group.
Relation with other modifiers
Stronger property modifiers
This group property modifier is monotone, viz if are group properties and is the operator, then
If are group properties, then .
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
For any group property , . In other words, if a group satisfies property it also satisfies property .
This group property modifier is idempotent, viz applying it twice to a group property has the same effect as applying it once
For any group property , . In other words, applying the poly operator twice has the same effect as applying it once.