Replacement property

From Groupprops
Revision as of 00:07, 8 May 2008 by Vipul (talk | contribs) (2 revisions)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]


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.


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.