# Replacement property

## Definition

Let and be two subgroup properties. A finite group is said to satisfy a *replacement property* if for any subgroup of satisfying property , there exists a subgroup of such that:

- satisfies property in
- has the same order as

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

## 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.