Replacement property

Definition
Let $$p$$ and $$q$$ be two subgroup properties. A finite group $$G$$ is said to satisfy a $$p \to q$$ replacement property if for any subgroup $$H$$ of $$G$$ satisfying property $$p$$, there exists a subgroup $$K$$ of $$G$$ such that:


 * $$K$$ satisfies property $$q$$ in $$G$$
 * $$K$$ has the same order as $$H$$

In most practical situations, we assume that $$q$$ is a stronger property than $$p$$.

Importance
Replacement theorems are theorems that prove replacement properties. The key goal of a replacement theorem is to provide a guarantee that we can pass from a subgroup satisfying a weaker set of constraints, to a subgroup satisfying a stronger set of constraints.