2-subnormal subgroup

Definition

QUICK PHRASES: normal inside normal closure, every conjugate is in its normalizer, normal closure is in normalizer, normal subgroup of normal subgroup, subgroup of subnormal defect at most two

No.	Shorthand	A subgroup of a group is 2-subnormal in it if ...	A subgroup $H$ of a group $G$ if 2-subnormal in $G$ if ...
1	Normal of normal	there is an intermediate subgroup containing it such that the subgroup is normal in the intermediate subgroup and such that the intermediate subgroup is normal in the whole group.	there is subgroup $K$ of $G$ such that $H$ is a normal subgroup of $K$ and $K$ is a normal subgroup of $G$.
2	Normal in closure	the subgroup is normal in its normal closure in the whole group.	$H$ is a normal subgroup of its normal closure $H^G$ in $G$.
3	Normal closure in normalizer	the normal closure of the subgroup is contained in the normalizer of the subgroup.	the normal closure $H^G$ is contained in the normalizer $N_G(H)$.
4	Every conjugate in normalizer	every conjugate of the subgroup is contained in its normalizer, i.e., every conjugate normalizes it.	for every $g \in G$, $gHg^{-1} \le N_G(H)$.
5	In normal core of normalizer	the subgroup is contained in its normal core of normalizer: the normal core of its normalizer.	$H$ is contained in the normal core $(N_G(H))_G$ of $N_G(H)$ in $G$.
6	Subnormal of depth 2	it is a subnormal subgroup whose subnormal depth (also called subnormal defect) is at most $2$.	$H$ is subnormal in $G$ with subnormal depth at most $2$.
7	contains second commutator	it contains its second commutator subgroup with the whole group	$[[G,H],H] \le H$ where $[,]$ denotes the commutator of two subgroups.

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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this: (facts closely related to 2-subnormal subgroup, all facts related to 2-subnormal subgroup) |Survey articles about this | Survey articles about definitions built on this
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View a complete list of semi-basic definitions on this wiki

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: normal subgroup and normal subgroup
View other such compositions|View all subgroup properties

This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

Comment

A 2-subnormal subgroup $H$ has a unique fastest ascending subnormal series $H \le K \le G$, where $K$ is the normal core of $N_G(H)$. It also has a unique fastest descending subnormal series $G \ge L \ge H$, where $L$ is the normal closure of $H$ in $G$. While subnormal subgroups of larger depth also have unique fastest descending subnormal series, they do not in general possess unique fastest ascending subnormal series. Further information: 2-subnormal subgroup has a unique fastest ascending subnormal series, Subnormal subgroup has a unique fastest descending subnormal series, 3-subnormal subgroup need not have a unique fastest ascending subnormal series

Formalisms

First-order description

This subgroup property is a first-order subgroup property, viz., it has a first-order description in the theory of groups.
View a complete list of first-order subgroup properties

A subgroup $H$ is 2-subnormal in a group $G$ if it satisfies the following first-order sentence:

$\forall g \in G, \forall x,y \in H, gxg^{-1}ygx^{-1}g^{-1} \in H$

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
Base of a wreath product	occurs as the base (the direct factor in the direct product that forms the normal subgroup of the semidirect product) of a wreath product			Base of a wreath product with diagonal action, Central factor of normal subgroup, Direct factor of complemented normal subgroup, Direct factor of normal subgroup, Right-transitively 2-subnormal subgroup, Transitively normal subgroup of normal subgroup|FULL LIST, MORE INFO
Normal subgroup	invariant under inner automorphisms; subnormal of depth at most 1	(by definition)	normality is not transitive	2-hypernormalized subgroup, Central factor of normal subgroup, Direct factor of normal subgroup, Join-transitively 2-subnormal subgroup, Modular 2-subnormal subgroup, Normal subgroup of characteristic subgroup, Permutable 2-subnormal subgroup, Transitively normal subgroup of normal subgroup|FULL LIST, MORE INFO
2-hypernormalized subgroup	normalizer is normal		2-subnormal not implies hypernormalized	|FULL LIST, MORE INFO
Right-transitively 2-subnormal subgroup	every 2-subnormal subgroup of it is 2-subnormal in the whole group.	(by definition, since 2-subnormality is an identity-true subgroup property)	2-subnormality is not transitive	|FULL LIST, MORE INFO
Left-transitively 2-subnormal subgroup	If whole group is 2-subnormal in some group, so is subgroup	(by definition, since 2-subnormality is an identity-true subgroup property)	2-subnormality is not transitive	Normal subgroup of characteristic subgroup|FULL LIST, MORE INFO
Join-transitively 2-subnormal subgroup	join with any 2-subnormal subgroup is 2-subnormal	(by definition, since 2-subnormality is a trivially true subgroup property)	2-subnormality is not finite-join-closed	|FULL LIST, MORE INFO
Commutator of a normal subgroup and a subset		Commutator of a normal subgroup and a subset implies 2-subnormal		|FULL LIST, MORE INFO
Direct factor of characteristic subgroup	direct factor of characteristic subgroup			Central factor of normal subgroup, Direct factor of normal subgroup, Normal subgroup of characteristic subgroup|FULL LIST, MORE INFO
Direct factor of normal subgroup	direct factor of a normal subgroup of the whole group			Central factor of normal subgroup, Transitively normal subgroup of normal subgroup|FULL LIST, MORE INFO
Normal subgroup of characteristic subgroup	normal subgroup of a characteristic subgroup of the whole group	follows from characteristic implies normal		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Conjugate-permutable subgroup	permutes with all conjugate subgroups	2-subnormal implies conjugate-permutable	conjugate-permutable not implies 2-subnormal	Permutable subgroup of normal subgroup|FULL LIST, MORE INFO
Join-transitively subnormal subgroup	join with any subnormal subgroup is subnormal	2-subnormal implies join-transitively subnormal	join-transitively subnormal not implies 2-subnormal	Conjugate-join-closed subnormal subgroup, Intermediately join-transitively subnormal subgroup, Join of finitely many 2-subnormal subgroups, Linear-bound join-transitively subnormal subgroup, Polynomial-bound join-transitively subnormal subgroup|FULL LIST, MORE INFO
Linear-bound join-transitively subnormal subgroup				|FULL LIST, MORE INFO
Polynomial-bound join-transitively subnormal subgroup				|FULL LIST, MORE INFO
Join of finitely many 2-subnormal subgroups			2-subnormality is not finite-join-closed	|FULL LIST, MORE INFO
Join of 2-subnormal subgroups				|FULL LIST, MORE INFO
Subnormal subgroup			there exist subgroups of arbitrarily large subnormal depth	3-subnormal subgroup, 4-subnormal subgroup, Finite-conjugate-join-closed subnormal subgroup, Intermediately join-transitively subnormal subgroup, Join of finitely many 2-subnormal subgroups, Join-transitively subnormal subgroup, Linear-bound join-transitively subnormal subgroup|FULL LIST, MORE INFO
3-subnormal subgroup	subnormal of depth at most 3			|FULL LIST, MORE INFO
4-subnormal subgroup	subnormal of depth at most 4			|FULL LIST, MORE INFO

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	No	2-subnormality is not transitive	There exist groups $H \le K \le G$, with $H$ 2-subnormal in $K$, $K$ 2-subnormal in $G$, but $H$ not 2-subnormal in $G$.
trim subgroup property	Yes	Every group is normal in itself, trivial subgroup is normal	For any group $G$, the whole group and the trivial subgroup are both 2-subnormal.
strongly intersection-closed subgroup property	Yes	Subnormality of fixed depth is strongly intersection-closed	$H_i, i \in I$ all 2-subnormal subgroups of $G$, then so is $\bigcap_{i \in I} H_i$.
finite-join-closed subgroup property	No	2-subnormality is not finite-join-closed	Can have subgroups $H, K \le G$, both 2-subnormal in $G$, such that $\langle H, K \rangle$ is not 2-subnormal.
conjugate-join-closed subgroup property	Yes	2-subnormality is conjugate-join-closed	A join of subgroups $H_i, i \in I$ of $G$, with all 2-subnormal in $G$ and all conjugate to each other, is also 2-subnormal.
intermediate subgroup condition	Yes	2-subnormality satisfies intermediate subgroup condition	If $H \le K \le G$, with $H$ 2-subnormal in $G$, then $H$ is 2-subnormal in $K$.
transfer condition	Yes	2-subnormality satisfies transfer condition	If $H, K \le G$, with $H$ 2-subnormal in $G$, then $H \cap K$ is 2-subnormal in $K$.
image condition	Yes	2-subnormality satisfies image condition	If $H$ 2-subnormal in $G$, $f:G \to K$ surjective homomorphism, then $f(H)$ is 2-subnormal in $K$.
inverse image condition	Yes	2-subnormality satisfies inverse image condition	If $f:K \to G$ homomorphism, $H$ 2-subnormal in $G$, then $f^{-1}(H)$ 2-subnormal in $K$.
upper join-closed subgroup property	No	2-subnormality is not upper join-closed	Can have $H \le K, L \le G$ with $H$ 2-subnormal in both $K$ and $L$ but not in $\langle K, L \rangle$.

For more details of these metaproperties:[SHOW MORE]

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

A 2-subnormal subgroup of a 2-subnormal subgroup is not necessarily 2-subnormal. For full proof, refer: 2-subnormality is not transitive

Further information: Normality is not transitive, there exist subgroups of arbitrarily large subnormal depth, normal not implies left-transitively fixed-depth subnormal, normal not implies right-transitively fixed-depth subnormal

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Every group is 2-subnormal as a subgroup of itself, and further, the trivial subgroup is 2-subnormal in any group.

This follows from the fact that every group is normal in itself and the trivial subgroup is also normal in every group. For full proof, refer: trivial subgroup is normal, every group is normal in itself

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

An arbitrary intersection of 2-subnormal subgroups is 2-subnormal. For full proof, refer: 2-subnormality is strongly intersection-closed

Join-closedness

This subgroup property is not join-closed, viz., it is not true that a join of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not join-closed

An arbitrary join of 2-subnormal subgroups need not be 2-subnormal. In fact, even a join of two 2-subnormal subgroups need not be 2-subnormal. For full proof, refer: 2-subnormality is not finite-join-closed

Conjugate-join-closedness

This subgroup property is conjugate-join-closed; in other words, a join of conjugate subgroups, each having the property, also has the property.
View a complete list of conjugate-join-closed subgroup properties

A join of 2-subnormal subgroups that are conjugate to each other is again 2-subnormal. For full proof, refer: 2-subnormality is conjugate-join-closed

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H$ is a 2-subnormal subgroup of $G$, then $H$ is also 2-subnormal in any intermediate subgroup $K$ of $G$. For full proof, refer: 2-subnormality satisfies intermediate subgroup condition

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

If $H$ is a 2-subnormal subgroup of $G$ and $K$ is any subgroup of $G$, then $H \cap K$ is 2-subnormal in $K$. For full proof, refer: 2-subnormality satisfies transfer condition

Image condition

YES: This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
View other subgroup properties satisfying image condition

If $f:G \to K$ is a surjective homomorphism of groups, and $H$ is 2-subnormal in $G$, then $f(H)$ is 2-subnormal in $K$.

Upper join-closedness

NO: This subgroup property is not upper join-closed: if a subgroup has the property in intermediate subgroups it need not have the property in their join.

If $H \le G$ and $K, L$ are two intermediate subgroups containing $H$, it may happen that $H$ is 2-subnormal in $K$ as well as in $L$, but is not 2-subnormal in $\langle K, L \rangle$. For full proof, refer: 2-subnormality is not upper join-closed

Effect of property operators

Operator	Meaning	Result of application	Proof
left transiter	if big group is 2-subnormal in a bigger group, so is subgroup	left-transitively 2-subnormal subgroup	by definition
right transiter	any 2-subnormal subgroup of subgroup is 2-subnormal in whole group	right-transitively 2-subnormal subgroup	by definition
join-transiter	join with any 2-subnormal subgroup is 2-subnormal	join-transitively 2-subnormal subgroup	by definition

For more information on these operators: [SHOW MORE]

The right transiter

Applying the right transiter to this property gives: right-transitively 2-subnormal subgroup

The right transiter of the property of being 2-subnormal is termed the property of being right-transitively 2-subnormal. A subgroup $H$ of a group $G$ is termed right-transitively 2-subnormal if any 2-subnormal subgroup $K$ of $H$ is 2-subnormal in $G$.

Some subgroup properties stronger than being right-transitively 2-subnormal include: base of a wreath product, transitively normal subgroup, and normal subgroup that is also a T-group (for instance, an Abelian normal subgroup).

The left transiter

Applying the left transiter to this property gives: left-transitively 2-subnormal subgroup

The left transiter of the property of being 2-subnormal is termed the property of being left-transitively 2-subnormal. A subgroup $H$ of a group $G$ is termed left-transitively 2-subnormal if whenever $G$ is embedded as a 2-subnormal subgroup of some group $K$, $H$ is also 2-subnormal in $K$.

Any characteristic subgroup is left-transitively 2-subnormal, because the left transiter of normal is characteristic.

The join-transiter

Applying the join-transiter to this property gives: join-transitively 2-subnormal subgroup

A join-transitively 2-subnormal subgroup is a subgroup whose join with any 2-subnormal subgroup is 2-subnormal. Any normal subgroup is join-transitively 2-subnormal.