Normal fully normalized subgroup
From Groupprops
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and fully normalized subgroup
View other subgroup property conjunctions | view all subgroup properties
Contents
Definition
Symbol-free definition
A subgroup of a group is termed normal fully normalized if it is normal in the whole group, and every automorphism of the subgroup extends to an inner automorphism of the whole group.
Definition with symbols
A subgroup of a group
is termed normal fully normalized if
is normal in
, and every
can be realized as conjugation by
for some
.
Equivalently, is normal in
, and the natural map
is surjective.
Relation with other properties
Conjunction with other properties
- NSCFN-subgroup is the conjunction with the property of being a self-centralizing subgroup.
Weaker properties
Facts
- Every group is normal fully normalized in its holomorph
- Normal upper-hook fully normalized implies characteristic
- Characteristically simple and normal fully normalized implies minimal normal
Metaproperties
Left realization
Every group can be realized as a normal fully normalized subgroup in some group, for instance, in its holomorph. Further information: Every group is normal fully normalized in its holomorph