Normal subgroup of characteristic subgroup
From Groupprops
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
Contents
Definition
Symbol-free definition
A subgroup of a group is termed a normal subgroup of characteristic subgroup if it satisfies the following equivalent conditions:
- It is a normal subgroup of a characteristic subgroup of the group.
- It is normal inside its characteristic closure in the group.
- Its characteristic closure is contained in its normalizer.
- It is contained in the characteristic core of its normalizer.
Definition with symbols
A subgroup of a group
is termed a normal subgroup of characteristic subgroup if it satisfies the following equivalent conditions:
- There exists a characteristic subgroup
of
such that
is a normal subgroup of
.
-
is a normal subgroup inside the characteristic closure of
in
.
- The characteristic closure of
in
is contained in the normalizer
.
-
is contained in the characteristic core of
in
.
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
is a subgroup of
with the property that whenever
is normal in a group
,
is 2-subnormal in
, then
is a normal subgroup of characteristic subgroup in
.
Effect of property operators
The left transiter
Applying the left transiter to this property gives: left-transitively 2-subnormal subgroup