Subordination operator

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

View a complete list of subgroup property modifiers OR View a list of all subgroup property operators (possibly with multiple inputs)

Definition

Symbol-free definition

The subordination operator is a map from the subgroup property space to itself that sends a subgroup property $p$ to the property of being a subgroup for which there exists a ascending chain of subgroups from the subgroup to the group with each member satisfying $p$ in its successor.

Definition with symbols

The subordination operator on a property $p$ gives the following property: $H$ satisfies it in $G$ if there is an ascending chain $H = H_0$ ≤ $H_1$ ≤ $... H_n = G$ with each $H_i$ satisfying $p$ in $H_{i+1}$.

Property theory of the subordination operator

Transitive and identity-true

As for a general Kleene star operator, the subordination operator is a monotone descendant operator and is also idempotent. The fixed points are precisely the transitive identity-true properties.