Normalizer-relatively normal subgroup

From Groupprops
Jump to: navigation, search
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article describes a property that can be evaluated for a triple of a group, a subgroup of the group, and a subgroup of that subgroup.
View other such properties

Definition

Suppose H \le K \le G are groups. We say that H is a normalizer-relatively normal subgroup of K with respect to G if the following equivalent conditions are satisfied:

  • Whenever K \le L \le G is such that K is normal in L, H is also normal in L.
  • The normalizer N_G(K) is contained in the normalizer N_G(H).
  • H is normal in N_G(K).

Relation with other properties

Stronger properties

Weaker properties