Normal subgroup of characteristic subgroup

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: normal subgroup and characteristic subgroup
View other such compositions|View all subgroup properties

Definition

Symbol-free definition

A subgroup of a group is termed a normal subgroup of characteristic subgroup if it satisfies the following equivalent conditions:

1. It is a normal subgroup of a characteristic subgroup of the group.
2. It is normal inside its characteristic closure in the group.
3. Its characteristic closure is contained in its normalizer.
4. It is contained in the characteristic core of its normalizer.

Definition with symbols

A subgroup $H$ of a group $G$ is termed a normal subgroup of characteristic subgroup if it satisfies the following equivalent conditions:

1. There exists a characteristic subgroup $K$ of $G$ such that $H$ is a normal subgroup of $K$.
2. $H$ is a normal subgroup inside the characteristic closure of $H$ in $G$.
3. The characteristic closure of $H$ in $G$ is contained in the normalizer $N_G(H)$.
4. $H$ is contained in the characteristic core of $N_G(H)$ in $G$.

Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup				|FULL LIST, MORE INFO
characteristic subgroup				Direct factor of characteristic subgroup, Left-transitively 2-subnormal subgroup|FULL LIST, MORE INFO
direct factor of characteristic subgroup				|FULL LIST, MORE INFO
left-transitively 2-subnormal subgroup		Left-transitively 2-subnormal implies normal of characteristic		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
automorph-permutable subgroup				|FULL LIST, MORE INFO
2-subnormal subgroup				|FULL LIST, MORE INFO
3-subnormal subgroup				2-subnormal subgroup|FULL LIST, MORE INFO
4-subnormal subgroup				2-subnormal subgroup|FULL LIST, MORE INFO
subnormal subgroup				2-subnormal subgroup|FULL LIST, MORE INFO

Facts

• Left residual of 2-subnormal by normal is normal of characteristic: If $H$ is a subgroup of $K$ with the property that whenever $K$ is normal in a group $G$, $H$ is 2-subnormal in $G$, then $H$ is a normal subgroup of characteristic subgroup in $K$.

Effect of property operators

The left transiter

Applying the left transiter to this property gives: left-transitively 2-subnormal subgroup