Transitively normal subgroup
From Groupprops
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
Definition
Equivalent definitions in tabular format
No. | Shorthand | A subgroup of a group is termed transitively normal if ... | A subgroup ![]() ![]() |
---|---|---|---|
1 | transitively normal | every normal subgroup of the subgroup is normal in the whole group. | whenever ![]() ![]() ![]() ![]() |
2 | normal automorphisms restriction | every normal automorphism of the whole group restricts to a normal automorphism of the subgroup. | for any normal automorphism ![]() ![]() ![]() ![]() ![]() |
3 | normal CEP-subgroup | it is a normal subgroup as well as a CEP-subgroup: every normal subgroup of it is an intersection with it of a normal subgroup of the whole group | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Equivalence of definitions
Further information: Equivalence of definitions of transitively normal subgroup
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 page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and CEP-subgroup
View other subgroup property conjunctions | view all subgroup properties
This is a variation of normality|Find other variations of normality | Read a survey article on varying normality
CAUTIONARY NOTE: There is a paper where the term transitively normal is used for what we call intermediately subnormal-to-normal subgroup
Examples
Extreme examples
- Every group is transitively normal as a subgroup of itself.
- The trivial subgroup is transitively normal in any group.
Examples in small finite groups
Below are some examples of a proper nontrivial subgroup that satisfy the property transitively normal subgroup.
Below are some examples of a proper nontrivial subgroup that does not satisfy the property transitively normal subgroup.
Metaproperties
BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
transitive subgroup property | Yes | transitive normality is transitive | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
intermediate subgroup condition | Yes | transitive normality satisfies intermediate subgroup condition | If ![]() ![]() ![]() ![]() ![]() |
finite-intersection-closed subgroup property | No | transitive normality is not finite-intersection-closed | It is possible to have a group ![]() ![]() ![]() ![]() ![]() |
finite-join-closed subgroup property | No | transitive normality is not finite-join-closed | It is possible to have a group ![]() ![]() ![]() ![]() ![]() |
centralizer-closed subgroup property | No | transitive normality is not centralizer-closed | It is possible to have a group ![]() ![]() ![]() ![]() |
image condition | Yes | transitive normality satisfies image condition | If ![]() ![]() ![]() ![]() ![]() |
quotient-transitive subgroup property | No | transitive normality is not quotient-transitive | It is possible to have groups ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Relation with other properties
Conjunction with other properties
Here are some conjunctions with other subgroup properties:
Conjunction | Other component of conjunction | Intermediate notions | Additional comments |
---|---|---|---|
characteristic transitively normal subgroup | characteristic subgroup | |FULL LIST, MORE INFO | |
c-closed transitively normal subgroup | c-closed subgroup (equals its double centralizer) | |FULL LIST, MORE INFO |
Conjunctions with group properties:
Conjunction | Property of the group | Intermediate notions | Additional comments |
---|---|---|---|
cyclic normal subgroup | Cyclic group | Abelian hereditarily normal subgroup, Hereditarily normal subgroup|FULL LIST, MORE INFO | for a cyclic subgroup, being normal is equivalent to being transitively normal. |
abelian hereditarily normal subgroup | Abelian group | Hereditarily normal subgroup|FULL LIST, MORE INFO | for an abelian subgroup, being transitively normal is equivalent to being hereditarily normal. |
hereditarily normal subgroup | Dedekind group (every subgroup is normal) | SCAB-subgroup|FULL LIST, MORE INFO | |
nilpotent transitively normal subgroup | Nilpotent group | |FULL LIST, MORE INFO |
Stronger properties
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
normal subgroup | (by definition) | normality is not transitive | |FULL LIST, MORE INFO | |
CEP-subgroup | every normal subgroup of it arises as its intersection with a normal subgroup of whole group | |FULL LIST, MORE INFO | ||
subgroup whose commutator with any subset is normal | its commutator with any subset of whole group is normal in whole group | commutator of a transitively normal subgroup and a subset implies normal | commutator with any subset is normal not implies transitively normal | |FULL LIST, MORE INFO |
Formalisms
BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
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 transitive normality is ... | Additional comments |
---|---|---|---|
normal automorphism ![]() |
every normal automorphism of ![]() ![]() |
the balanced subgroup property for normal automorphisms | Hence, it is a t.i. subgroup property, both transitive and identity-true |
inner automorphism ![]() |
every inner automorphism of ![]() ![]() |
||
class-preserving automorphism ![]() |
every class-preserving automorphism of ![]() ![]() |
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subgroup-conjugating automorphism ![]() |
every subgroup-conjugating automorphism of ![]() ![]() |
||
strong monomial automorphism ![]() |
every strong monomial automorphism of ![]() ![]() |