Poly operator

Property-theoretic definition
The poly operator takes as input a group property $$p$$ and outputs the Kleene star closure of $$p$$ with respect to the extension operator, bracketed on the left.

Definition with symbols
Given a group property $$p$$, the poly operator gives the group property $$q$$ defined as follows:

A group $$G$$ has property $$q$$ if we can find a subnormal series $$e = H_0 \triangleleft H_1 \triangleleft \ldots \triangleleft H_r = G$$ such that each $$H_r/H_{r-1}$$ satisfies property $$p$$ as an abstract group.

Stronger property modifiers

 * Finite normal series operator
 * Finite characteristic series operator

Properties
If $$p \le q$$ are group properties, then $$poly(p) \le poly(q)$$.

For any group property $$p$$, $$p \le poly(p)$$. In other words, if a group satisfies property $$p$$ it also satisfies property $$poly(p)$$.

For any group property $$p$$, $$poly(poly(p)) = poly(p)$$. In other words, applying the poly operator twice has the same effect as applying it once.