Finite normal series operator

Property-theoretic definition
The finite normal series operator is a group property modifier that takes as input a group property $$p$$ and outputs the Kleene star closure of $$p$$ with respect to the group extension operator, right-bracketed.

Definition with symbols
Given a group property $$p$$, the finite normal series operator applied to $$p$$ returns the group property $$q$$ defined as follows: a group $$G$$ has property $$q$$ if there is a finite normal series $$e = H_0 \le H_1 \le \ldots \le H_r = G$$ such that each $$H_i$$ is normal in $$G$$ and such that $$H_i/H_{i-1}$$ satisfies property $$p$$ for every $$i$$.

Stronger modifiers

 * Finite characteristic series operator

Weaker modifiers

 * Poly operator

Metaproperties
If $$p \le q$$ are group properties, then the image of the finite normal series operator on $$p$$ implies the image of the finite normal series operator on $$q$$.

For any group property $$p$$, any group satisfying $$p$$ also satisfies the image of $$p$$ under the finite normal series operator.

Idempotence
The finite normal series operator need not be idempotent.