Transitively normal subgroup

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Definition

Equivalent definitions in tabular format

No.	Shorthand	A subgroup of a group is termed transitively normal if ...	A subgroup $H$ of a group $G$ is termed transitively normal if ...
1	transitively normal	every normal subgroup of the subgroup is normal in the whole group.	whenever $K$ is a normal subgroup of $H$, $K$ is also a normal subgroup of $G$.
2	normal automorphisms restriction	every normal automorphism of the whole group restricts to a normal automorphism of the subgroup.	for any normal automorphism $\sigma$ of $G$, the restriction of $\sigma$ to $H$ is a normal automorphism of $H$.
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	$H$ is a normal subgroup of $G$ and for any normal subgroup $K$ of $H$ there exists a normal subgroup $L$ of $G$ such that $K = H \cap L$.

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]

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and CEP-subgroup
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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.

	Group part	Subgroup part	Quotient part
A3 in S3	Symmetric group:S3	Cyclic group:Z3	Cyclic group:Z2
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group
Center of quaternion group	Quaternion group	Cyclic group:Z2	Klein four-group
Cyclic maximal subgroup of dihedral group:D8	Dihedral group:D8	Cyclic group:Z4	Cyclic group:Z2
Derived subgroup of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Direct product of Z4 and Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Cyclic group:Z4
Q8 in central product of D8 and Z4	Central product of D8 and Z4	Quaternion group	Cyclic group:Z2
SL(2,3) in GL(2,3)	General linear group:GL(2,3)	Special linear group:SL(2,3)	Cyclic group:Z2
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Dihedral group:D8
Z2 in V4	Klein four-group	Cyclic group:Z2	Cyclic group:Z2
Z4 in direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z4	Cyclic group:Z2

Below are some examples of a proper nontrivial subgroup that does not satisfy the property transitively normal subgroup.

	Group part	Subgroup part	Quotient part
Klein four-subgroups of dihedral group:D8	Dihedral group:D8	Klein four-group	Cyclic group:Z2

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 $H \le K \le G$ are groups such that $H$ is transitively normal in $K$ and $K$ is transitively normal in $G$, then $H$ is transitively normal in $G$.
intermediate subgroup condition	Yes	transitive normality satisfies intermediate subgroup condition	If $H \le K \le G$ and $H$ is transitively normal in $G$, then $H$ is transitively normal in $K$.
finite-intersection-closed subgroup property	No	transitive normality is not finite-intersection-closed	It is possible to have a group $G$ and transitively normal subgroups $H_1,H_2$ of $G$ such that $H_1 \cap H_2$ is not transitively normal in $G$.
finite-join-closed subgroup property	No	transitive normality is not finite-join-closed	It is possible to have a group $G$ and transitively normal subgroups $H_1,H_2$ of $G$ such that the join $\langle H_1,H_2\rangle$ (which is the same as the product $H_1H_2$) is not transitively normal.
centralizer-closed subgroup property	No	transitive normality is not centralizer-closed	It is possible to have a group $G$ and a transitively normal subgroup $H$ of $G$ such that the centralizer $C_G(H)$ is not transitively normal.
image condition	Yes	transitive normality satisfies image condition	If $H$ is a transitively normal subgroup of $G$ and $\varphi:G \to K$ is a surjective homomorphism, then $\varphi(H)$ is a transitively normal subgroup of $K$.
quotient-transitive subgroup property	No	transitive normality is not quotient-transitive	It is possible to have groups $H \le K \le G$ such that $H$ is transitively normal in $G$ and $K/H$ is transitively normal in $G/H$, but $K$ is not transitively normal in $G$.

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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
central factor	product with centralizer is whole group	central factor implies transitively normal	transitively normal not implies central factor (see also list of examples)	Conjugacy-closed normal subgroup, Locally inner automorphism-balanced subgroup, SCAB-subgroup|FULL LIST, MORE INFO
direct factor	factor in an internal direct product	direct factor implies transitively normal (also, via central factor)	(via central factor) (see also list of examples)	Central factor, Complemented transitively normal subgroup, Conjugacy-closed normal subgroup, Join of finitely many direct factors, Join-transitively central factor, Locally inner automorphism-balanced subgroup, Right-quotient-transitively central factor, SCAB-subgroup, Upper join of direct factors|FULL LIST, MORE INFO
cocentral subgroup	product with center is whole group	(via central factor)	(via central factor)	Central factor, Conjugacy-closed normal subgroup, Right-quotient-transitively central factor, SCAB-subgroup|FULL LIST, MORE INFO
central subgroup	contained in the center	(via central factor, via hereditarily normal)	(via central factor, via hereditarily normal)	Abelian hereditarily normal subgroup, Central factor, Conjugacy-closed normal subgroup, Hereditarily normal subgroup, Join-transitively central factor, SCAB-subgroup|FULL LIST, MORE INFO
locally inner autorphism-balanced subgroup	every inner automorphism of the whole group restricts to a locally inner automorphism			|FULL LIST, MORE INFO
conjugacy-closed normal subgroup	both conjugacy-closed and normal	conjugacy-closed normal implies transitively normal	transitively normal not implies conjugacy-closed	|FULL LIST, MORE INFO
SCAB-subgroup	any subgroup-conjugating automorphism of whole group restricts to subgroup-conjugating automorphism of subgroup	SCAB implies transitively normal	Transitively normal not implies SCAB	|FULL LIST, MORE INFO
hereditarily normal subgroup	every subgroup of it is normal in whole group			SCAB-subgroup|FULL LIST, MORE INFO
cyclic normal subgroup	cyclic and normal	(via SCAB, via hereditarily normal)	(via SCAB, via hereditarily normal)	Abelian hereditarily normal subgroup, Hereditarily normal subgroup|FULL LIST, MORE INFO

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	$H$ is a transitively normal subgroup of $G$ if ...	This means that transitive normality is ...	Additional comments
normal automorphism $\to$ normal automorphism	every normal automorphism of $G$ restricts to a normal automorphism of $H$	the balanced subgroup property for normal automorphisms	Hence, it is a t.i. subgroup property, both transitive and identity-true
inner automorphism $\to$ normal automorphism	every inner automorphism of $G$ restricts to a normal automorphism of $H$
class-preserving automorphism $\to$ normal automorphism	every class-preserving automorphism of $G$ restricts to a normal automorphism of $H$
subgroup-conjugating automorphism $\to$ normal automorphism	every subgroup-conjugating automorphism of $G$ restricts to a normal automorphism of $H$
strong monomial automorphism $\to$ normal automorphism	every strong monomial automorphism of $G$ restricts to a normal automorphism of $H$