Hereditarily normal subgroup
This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This article describes a property that arises as the conjunction of a subgroup property: transitively normal subgroup with a group property (itself viewed as a subgroup property): Dedekind group
View a complete list of such conjunctions
Definition
Symbol-free definition
A subgroup of a group is termed hereditarily normal or quasicentral if it satisfies the following equivalent conditions:
- Every subgroup of the subgroup is a normal subgroup of the whole group.
- It is both a transitively normal subgroup of the whole group and a Dedekind group.
- All inner automorphisms of the whole group restrict to power automorphisms on the subgroup.
Definition with symbols
A subgroup of a group
is termed hereditarily normal or quasicentral if it satisfies the following equivalent conditions:
- For every subgroup
of
,
is a normal subgroup of
.
-
is both a Dedekind group and a transitively normal subgroup of
.
- For any
, the inner automorphism of
given by conjugation by
restricts to a power automorphism of
.
Formalisms
In terms of the hereditarily operator
This property is obtained by applying the hereditarily operator to the property: normal subgroup
View other properties obtained by applying the hereditarily operator
The property of being hereditarily normal is a result of applying the hereditarily operator on the property of normality.
Function restriction expression
This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
Function restriction expression | ![]() ![]() |
This means that "hereditarily normal" is ... | Additional comments |
---|---|---|---|
inner automorphism ![]() |
every inner automorphism of ![]() ![]() |
||
class-preserving automorphism ![]() |
every class-preserving automorphism of ![]() ![]() |
||
subgroup-conjugating automorphism ![]() |
every subgroup-conjugating automorphism of ![]() ![]() |
||
normal automorphism ![]() |
every normal automorphism of ![]() ![]() |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Central subgroup | contained in the center | central implies hereditarily normal | hereditarily normal not implies central | Abelian hereditarily normal subgroup|FULL LIST, MORE INFO |
Cyclic normal subgroup | normal and cyclic as a group | Abelian hereditarily normal subgroup|FULL LIST, MORE INFO | ||
Abelian hereditarily normal subgroup | abelian subgroup and every subgroup of it is normal in whole group | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
SCAB-subgroup | every subgroup-conjugating automorphism of the whole group restricts to a subgroup-conjugating automorphism of the subgroup | hereditarily normal implies SCAB | SCAB not implies hereditarily normal | |FULL LIST, MORE INFO |
Transitively normal subgroup | every normal subgroup of it is normal in the whole group | SCAB-subgroup|FULL LIST, MORE INFO | ||
Hereditarily permutable subgroup | every subgroup of it is permutable in the whole group | |FULL LIST, MORE INFO | ||
Hereditarily subnormal subgroup | every subgroup of it is subnormal in the whole group | Nilpotent subnormal subgroup|FULL LIST, MORE INFO | ||
Hereditarily pronormal subgroup | every subgroup of it is pronormal in the whole group | |FULL LIST, MORE INFO | ||
Normal subgroup | Dedekind normal subgroup, SCAB-subgroup, Transitively normal subgroup|FULL LIST, MORE INFO |
Metaproperties
Left-hereditariness
This subgroup property is left-hereditary: any subgroup of a subgroup with this property also has this property. Hence, it is also a transitive subgroup property.
Since the left-hereditarily operator is idempotent, the property of being hereditarily normal is itself left hereditary (that is, every subgroup of a hereditarily normal subgroup is hereditarily normal).
Note that being a left-hereditary property, it is automatically transitive and also intersection-closed.
Transfer condition
YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
View other subgroup properties satisfying the transfer condition
Since normality satisfies the intermediate subgroup condition, and the left-hereditarily operator preserves the intermediate subgroup condition, the property of being hereditarily normal also satisfies the transfer. condition. Hence it also satisfies the intermediate subgroup condition.
Trimness
The trivial group is obviously hereditarily normal.
The whole group is hereditarily normal if and only if every subgroup of the group is normal. Groups with this property are either Abelian groups or Hamiltonian groups.