Replacement property

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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.