Fully normalized potentially fully invariant subgroup

From Groupprops
Jump to: navigation, search
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: fully normalized subgroup and potentially fully invariant subgroup
View other subgroup property conjunctions | view all subgroup properties

Definition

A subgroup H of a group G is termed a fully normalized potentially fully invariant subgroup of G if it satisfies both these conditions:

  1. H is a fully normalized subgroup of G, i.e., every automorphism of H arises as the restriction to H of an inner automorphism of G.
  2. H is a potentially fully invariant subgroup of G, i.e., there exists a group K containing G such that H is a fully invariant subgroup of K.

Relation with other properties

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
centralizer-annihilating endomorphism-invariant subgroup fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant
normal fully normalized subgroup follows from potentially fully invariant implies normal