# 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
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## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup Direct factor of characteristic subgroup, Left-transitively 2-subnormal 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
• Left residual of 2-subnormal by normal is normal of characteristic: If $H$ is a subgroup of $G$ with the property that whenever $G$ is normal in a group $L$, $H$ is 2-subnormal in $L$, then $H$ is a normal subgroup of characteristic subgroup in $G$.