2-subnormal subgroup

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Definition

QUICK PHRASES: normal inside normal closure, every conjugate is in its normalizer, normal closure is in normalizer, normal subgroup of normal subgroup

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.

This article is about a standard (though not very rudimentary) definition in group theory.[SHOW MORE]
The article text may, however, contain more than just the basic definition
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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

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Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Base of a wreath product				click here
Normal subgroup		(by definition)	normality is not transitive	click here
2-hypernormalized subgroup	normalizer is normal		2-subnormal not implies hypernormalized
Right-transitively 2-subnormal subgroup	every 2-subnormal subgroup of it is 2-subnormal in the whole group.
Left-transitively 2-subnormal subgroup	If whole group is 2-subnormal in some group, so is subgroup			click here
Join-transitively 2-subnormal subgroup	join with any 2-subnormal subgroup is 2-subnormal		2-subnormality is not finite-join-closed
Commutator of a normal subgroup and a subset		Commutator of a normal subgroup and a subset implies 2-subnormal
Direct factor of characteristic subgroup				click here
Direct factor of normal subgroup				click here
Normal subgroup of characteristic subgroup

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	click here
Join-transitively subnormal subgroup	join with any subnormal subgroup is subnormal	2-subnormal implies join-transitively subnormal	join-transitively subnormal not implies 2-subnormal	click here
Linear-bound join-transitively subnormal subgroup
Polynomial-bound join-transitively subnormal subgroup
Join of finitely many 2-subnormal subgroups			2-subnormality is not finite-join-closed
Join of 2-subnormal subgroups
Subnormal subgroup			there exist subgroups of arbitrarily large subnormal depth	click here
3-subnormal subgroup	subnormal of depth at most 3
4-subnormal subgroup	subnormal of depth at most 4

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]

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]

Facts about 2-subnormal subgroupRDF feed

Applying operator gives	Right-transitively 2-subnormal subgroup +, Left-transitively 2-subnormal subgroup +, and Join-transitively 2-subnormal subgroup +
Defining ingredient	Normal subgroup +, Normal closure +, Normalizer of a subgroup +, Conjugate subgroups +, Normal core of normalizer +, Normal core +, Subnormal subgroup +, Subnormal depth +, and Commutator of two subgroups +
Dissatisfies metaproperty	Transitive subgroup property +, Join-closed subgroup property +, Finite-join-closed subgroup property +, and Upper join-closed subgroup property +
Page class	Term +
Satisfies metaproperty	Trim subgroup property +, Trivially true subgroup property +, Identity-true subgroup property +, Left-realized subgroup property +, Right-realized subgroup property +, Intersection-closed subgroup property +, Conjugate-join-closed subgroup property +, Intermediate subgroup condition +, Transfer condition +, Image condition +, First-order subgroup property +, Strongly intersection-closed subgroup property +, and Inverse image condition +
Stronger than	Conjugate-permutable subgroup +, Join-transitively subnormal subgroup +, Linear-bound join-transitively subnormal subgroup +, Polynomial-bound join-transitively subnormal subgroup +, Join of finitely many 2-subnormal subgroups +, Join of 2-subnormal subgroups +, Subnormal subgroup +, 3-subnormal subgroup +, and 4-subnormal subgroup +
Variation of	Normal subgroup +
Weaker than	Base of a wreath product +, Normal subgroup +, 2-hypernormalized subgroup +, Right-transitively 2-subnormal subgroup +, Left-transitively 2-subnormal subgroup +, Join-transitively 2-subnormal subgroup +, Commutator of a normal subgroup and a subset +, Direct factor of characteristic subgroup +, Direct factor of normal subgroup +, and Normal subgroup of characteristic subgroup +

2-subnormal subgroup

From Groupprops

Definition

Contents

Comment

Formalisms

First-order description

Examples

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Transitivity

Trimness

Intersection-closedness

Join-closedness

Conjugate-join-closedness

Intermediate subgroup condition

Transfer condition

Image condition

Upper join-closedness

Effect of property operators

The right transiter

The left transiter

The join-transiter

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