2-subnormal subgroup: Difference between revisions

Latest revision as of 04:46, 2 September 2026

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 $g \in G$ , $g H g^{- 1} \leq N_{G} (H)$ .
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	$[[G, H], H] \leq H$ where $[,]$ denotes the commutator of two subgroups.

This definition is presented using a tabular format. |View all pages with definitions in tabular format

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
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
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 $H \leq K \leq G$ , where $K$ is the normal core of $N_{G} (H)$ . It also has a unique fastest descending subnormal series $G \geq L \geq H$ , 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:

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

Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	No	2-subnormality is not transitive	There exist groups $H \leq K \leq G$ , 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 $⋂_{i \in I} H_{i}$ .
finite-join-closed subgroup property	No	2-subnormality is not finite-join-closed	Can have subgroups $H, K \leq G$ , both 2-subnormal in $G$ , such that $⟨ H, K ⟩$ 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 $H \leq K \leq G$ , with $H$ 2-subnormal in $G$ , then $H$ is 2-subnormal in $K$ .
transfer condition	Yes	2-subnormality satisfies transfer condition	If $H, K \leq G$ , 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 $H \leq K, L \leq G$ with $H$ 2-subnormal in both $K$ and $L$ but not in $⟨ K, L ⟩$ .

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			\|FULL LIST, MORE INFO
normal subgroup	invariant under inner automorphisms; subnormal of depth at most 1	(by definition)	normality is not transitive	\|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	\|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			\|FULL LIST, MORE INFO
direct factor of normal subgroup	direct factor of a normal subgroup of the whole group			\|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	\|FULL LIST, MORE INFO
loin-transitively subnormal subgroup	join with any subnormal subgroup is subnormal	2-subnormal implies join-transitively subnormal	join-transitively subnormal not implies 2-subnormal	\|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	\|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

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.

@@ Line 1: / Line 1: @@
+==Definition==
+{{quick phrase|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}}
+{| class="sortable" border="1"
+! No. !! Shorthand !! A subgroup of a group is 2-subnormal in it if ... !! A subgroup <math>H</math> of a group <math>G</math> if 2-subnormal in <math>G</math> if ...
+|-
+| 1 || normal of normal || there is an intermediate subgroup containing it such that the subgroup is [[defining ingredient::normal subgroup|normal]] in the intermediate subgroup and such that the intermediate subgroup is normal in the whole group. || there is subgroup <math>K</math> of <math>G</math> such that <math>H</math> is a normal subgroup of <math>K</math> and <math>K</math> is a normal subgroup of <math>G</math>.
+|-
+| 2 || normal in closure || the subgroup is normal in its [[defining ingredient::normal closure]] in the whole group. || <math>H</math> is a normal subgroup of its [[normal closure]] <math>H^G</math> in <math>G</math>.
+|-
+| 3 || normal closure in normalizer || the normal closure of the subgroup is contained in the [[defining ingredient::normalizer of a subgroup|normalizer]] of the subgroup. || the normal closure <math>H^G</math> is contained in the normalizer <math>N_G(H)</math>.
+|-
+| 4 || every conjugate in normalizer || every [[defining ingredient::conjugate subgroups|conjugate]] of the subgroup is contained in its normalizer, i.e., every conjugate normalizes it. || for every <math>g \in G</math>, <math>gHg^{-1} \le N_G(H)</math>.
+|-
+| 5 || in normal core of normalizer || the subgroup is contained in its [[defining ingredient::normal core of normalizer]]: the [[defining ingredient::normal core]] of its [[normalizer of a subgroup|normalizer]]. || <math>H</math> is contained in the normal core <math>(N_G(H))_G</math> of <math>N_G(H)</math> in <math>G</math>.
+|-
+| 6 || subnormal of depth 2 || it is a [[defining ingredient::subnormal subgroup]] whose [[defining ingredient::subnormal depth]] (also called ''subnormal defect'') is at most <math>2</math>. || <math>H</math> is subnormal in <math>G</math> with subnormal depth at most <math>2</math>.
+|-
+| 7 || contains second commutator || it contains its second commutator subgroup with the whole group || <math>[[G,H],H] \le H</math> where <math>[,]</math> denotes the [[defining ingredient::commutator of two subgroups]].
+|}
+{{tabular definition format}}
 {{semibasicdef}}
 {{subgroup property composition|normal subgroup|normal subgroup}}
 {{variation of|normality}}
-==Definition==
-{{quick phrase|normal inside normal closure, every conjugate is in its normalizer, normal closure is in normalizer, normal subgroup of normal subgroup}}
-===Symbol-free definition===
-A subgroup of a group is termed '''2-subnormal''' if the following equivalent conditions hold:
-* There is an intermediate subgroup containing it such that the subgroup is [[normal subgroup|normal]] in the intermediate subgroup and such that the intermediate subgroup is normal in the whole group.
-* The subgroup is normal in its [[normal closure]].
-* The [[normal closure]] of the subgroup is contained in its [[normalizer]]
-* The subgroup is contained in the [[normal core]] of its [[normalizer]]
-The property of being 2-subnormal is the same as the property of being subnormal of depth 2.
-===Definition with symbols===
-A subgroup <math>H</math> of a group <math>G</math> is termed '''2-subnormal''' if the following equivalent conditions hold:
+==Comment==
-* There is subgroup <math>K</math> such that <math>H</math> is a normal subgroup of <math>K</math> and <math>K</math> is a normal subgroup of <math>G</math>.
+A 2-subnormal subgroup <math>H</math> has a unique fastest ascending subnormal series <math>H \le K \le G</math>, where <math>K</math> is the normal core of <math>N_G(H)</math>. It also has a unique fastest descending subnormal series <math>G \ge L \ge H</math>, where <math>L</math> is the normal closure of <math>H</math> in <math>G</math>. 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|[[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]]}}
-* The normal closure of <math>H</math> is a normal subgroup of <math>G</math>.
 ==Formalisms==
@@ Line 31: / Line 37: @@
 <math>\forall g \in G, \forall x,y \in H, gxg^{-1}ygx^{-1}g^{-1} \in H</math>
-==Relation with other properties==
+==Examples==
-===Stronger properties===
-* [[Weaker than::Base of a wreath product]]
-* [[Weaker than::Normal subgroup]]: This follows directly from the definition. The strictness of the implication follows from the fact that [[normality is not transitive]]
-* [[Weaker than::2-hypernormalized subgroup]]: This is a particular case of the fact that any <math>k</math>-hypernormalized subgroup is also <math>k</math>-subnormal.
-* [[Weaker than::Right-transitively 2-subnormal subgroup]]
-* [[Weaker than::Left-transitively 2-subnormal subgroup]]
-* [[Weaker than::Direct factor of characteristic subgroup]]
-* [[Weaker than::Direct factor of normal subgroup]]
-* [[Weaker than::Normal subgroup of characteristic subgroup]]
-* [[Weaker than::Join-transitively 2-subnormal subgroup]]
-* [[Weaker than::Commutator of a normal subgroup and a subset]]: {{further|[[Commutator of a normal subgroup and a subset implies 2-subnormal]]}}
-===Weaker properties===
-* [[Stronger than::Conjugate-permutable subgroup]]: {{proofat|[[2-subnormal implies conjugate-permutable]]}}
+{{subgroup property see examples}}
-* [[Stronger than::Subnormal subgroup]]: This follows directly from the definition.
-* [[Stronger than::Join of finitely many 2-subnormal subgroups]]
-* [[Stronger than::Join of 2-subnormal subgroups]]
-* [[Stronger than::3-subnormal subgroup]]
-* [[Stronger than::Join-transitively subnormal subgroup]]: {{proofat|[[2-subnormal implies join-transitively subnormal]]}}
 ==Metaproperties==
-{{intransitive}}
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[dissatisfies metaproperty::transitive subgroup property]] || No || [[2-subnormality is not transitive]] || There exist groups <math>H \le K \le G</math>, with <math>H</math> 2-subnormal in <math>K</math>, <math>K</math> 2-subnormal in <math>G</math>, but <math>H</math> not 2-subnormal in <math>G</math>.
+|-
+| [[satisfies metaproperty::trim subgroup property]] || Yes || [[Every group is normal in itself]], [[trivial subgroup is normal]] || For any group <math>G</math>, the whole group and the trivial subgroup are both 2-subnormal.
+|-
+| [[satisfies metaproperty::strongly intersection-closed subgroup property]] || Yes || [[Subnormality of fixed depth is strongly intersection-closed]] || <math>H_i, i \in I</math> all 2-subnormal subgroups of <math>G</math>, then so is <math>\bigcap_{i \in I} H_i</math>.
+|-
+| [[dissatisfies metaproperty::finite-join-closed subgroup property]] || No || [[2-subnormality is not finite-join-closed]] || Can have subgroups <math>H, K \le G</math>, both 2-subnormal in <math>G</math>, such that <math>\langle H, K \rangle</math> is not 2-subnormal.
+|-
+| [[satisfies metaproperty::conjugate-join-closed subgroup property]] || Yes || [[2-subnormality is conjugate-join-closed]] || A join of subgroups <math>H_i, i \in I</math> of <math>G</math>, with all 2-subnormal in <math>G</math> and all conjugate to each other, is also 2-subnormal.
+|-
+| [[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[2-subnormality satisfies intermediate subgroup condition]] || If <math>H \le K \le G</math>, with <math>H</math> 2-subnormal in <math>G</math>, then <math>H</math> is 2-subnormal in <math>K</math>.
+|-
+| [[satisfies metaproperty::transfer condition]] || Yes || [[2-subnormality satisfies transfer condition]] || If <math>H, K \le G</math>, with <math>H</math> 2-subnormal in <math>G</math>, then <math>H \cap K</math> is 2-subnormal in <math>K</math>.
+|-
+| [[satisfies metaproperty::image condition]] || Yes || [[2-subnormality satisfies image condition]] || If <math>H</math> 2-subnormal in <math>G</math>, <math>f:G \to K</math> surjective homomorphism, then <math>f(H)</math> is 2-subnormal in <math>K</math>.
+|-
+| [[satisfies metaproperty::inverse image condition]] || Yes || [[2-subnormality satisfies inverse image condition]] || If <math>f:K \to G</math> homomorphism, <math>H</math> 2-subnormal in <math>G</math>, then <math>f^{-1}(H)</math> 2-subnormal in <math>K</math>.
+|-
+| [[dissatisfies metaproperty::upper join-closed subgroup property]] || No || [[2-subnormality is not upper join-closed]] || Can have <math>H \le K, L \le G</math> with <math>H</math> 2-subnormal in both <math>K</math> and <math>L</math> but not in <math>\langle K, L \rangle</math>.
+|}
-A 2-subnormal subgroup of a 2-subnormal subgroup is not necessarily 2-subnormal. {{proofat|[[2-subnormality is not transitive]]}}
+==Relation with other properties==
-{{trim}}
+===Stronger properties===
-Every group is 2-subnormal as a subgroup of itself, and further, the trivial subgroup is 2-subnormal in any group.
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::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 || || || {{intermediate notions short|2-subnormal subgroup|base of a wreath product}}
+|-
+| [[Weaker than::normal subgroup]] || invariant under inner automorphisms; subnormal of depth at most 1 || (by definition) || [[normality is not transitive]] || {{intermediate notions short|2-subnormal subgroup|normal subgroup}}
+|-
+| [[Weaker than::2-hypernormalized subgroup]] || [[normalizer]] is [[normal subgroup|normal]] || || [[2-subnormal not implies hypernormalized]] || {{intermediate notions short|2-subnormal subgroup|2-hypernormalized subgroup}}
+|-
+| [[Weaker than::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]] || {{intermediate notions short|2-subnormal subgroup|right-transitively 2-subnormal subgroup}}
+|-
+| [[Weaker than::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]]|| {{intermediate notions short|2-subnormal subgroup|left-transitively 2-subnormal subgroup}}
+|-
+| [[Weaker than::join-transitively 2-subnormal subgroup]] || [[join of subgroups|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]] || {{intermediate notions short|2-subnormal subgroup|join-transitively 2-subnormal subgroup}}
+|-
+| [[Weaker than::commutator of a normal subgroup and a subset]] || ||[[Commutator of a normal subgroup and a subset implies 2-subnormal]]|| || {{intermediate notions short|2-subnormal subgroup|commutator of a normal subgroup and a subset}}
+|-
+| [[Weaker than::direct factor of characteristic subgroup]] || [[direct factor]] of [[characteristic subgroup]]|| || || {{intermediate notions short|2-subnormal subgroup|direct factor of characteristic subgroup}}
+|-
+| [[Weaker than::direct factor of normal subgroup]] || [[direct factor]] of a [[normal subgroup]] of the whole group || || || {{intermediate notions short|2-subnormal subgroup|direct factor of normal subgroup}}
+|-
+| [[Weaker than::normal subgroup of characteristic subgroup]] || [[normal subgroup]] of a [[characteristic subgroup]] of the whole group || follows from [[characteristic implies normal]] || || {{intermediate notions short|2-subnormal subgroup|normal subgroup of characteristic subgroup}}
+|}
-{{intersection-closed}}
+===Weaker properties===
-An arbitrary intersection of 2-subnormal subgroups is 2-subnormal. {{proofat|[[2-subnormality is strongly intersection-closed]]}}
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::conjugate-permutable subgroup]] || [[permuting subgroups|permutes]] with all [[conjugate subgroups]] || [[2-subnormal implies conjugate-permutable]] || [[conjugate-permutable not implies 2-subnormal]] || {{intermediate notions short|conjugate-permutable subgroup|2-subnormal subgroup}}
+|-
+| [[Stronger than::loin-transitively subnormal subgroup]] || [[join of subgroups|join]] with any subnormal subgroup is subnormal || [[2-subnormal implies join-transitively subnormal]] || [[join-transitively subnormal not implies 2-subnormal]] || {{intermediate notions short|join-transitively subnormal subgroup|2-subnormal subgroup}}
+|-
+| [[Stronger than::linear-bound join-transitively subnormal subgroup]] || || || || {{intermediate notions short|linear-bound join-transitively subnormal subgroup|2-subnormal subgroup}}
+|-
+| [[Stronger than::polynomial-bound join-transitively subnormal subgroup]] || || || || {{intermediate notions short|polynomial-bound join-transitive subnormal subgroup|2-subnormal subgroup}}
+|-
+| [[Stronger than::join of finitely many 2-subnormal subgroups]] || || || [[2-subnormality is not finite-join-closed]] || {{intermediate notions short|join of finitely many 2-subnormal subgroups|2-subnormal subgroup}}
+|-
+| [[Stronger than::join of 2-subnormal subgroups]] || || || || {{intermediate notions short|join of 2-subnormal subgroups|2-subnormal subgroup}}
+|-
+| [[Stronger than::subnormal subgroup]] || || || [[there exist subgroups of arbitrarily large subnormal depth]] || {{intermediate notions short|subnormal subgroup|2-subnormal subgroup}}
+|-
+|[[Stronger than::3-subnormal subgroup]] || subnormal of depth at most 3|| || || {{intermediate notions short|3-subnormal subgroup|2-subnormal subgroup}}
+|-
+|[[Stronger than::4-subnormal subgroup]] || subnormal of depth at most 4 || || || {{intermediate notions short|4-subnormal subgroup|2-subnormal subgroup}}
+|}
-{{not join-closed}}
+==Effect of property operators==
-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. {{proofat|[[2-subnormality is not finite-join-closed]]}}
-{{conjugate-join-closed}}
-A join of 2-subnormal subgroups that are conjugate to each other is again 2-subnormal. {{proofat|[[2-subnormality is conjugate-join-closed]]}}
-{{intsubcondn}}
-If <math>H</math> is a 2-subnormal subgroup of <math>G</math>, then <math>H</math> is also 2-subnormal in any intermediate subgroup <math>K</math> of <math>G</math>. {{proofat|[[2-subnormality satisfies intermediate subgroup condition]]}}
-{{transfercondn}}
+{| class="sortable" border="1"
+! 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
+|}
-If <math>H</math> is a 2-subnormal subgroup of <math>G</math> and <math>K</math> is ''any'' subgroup of <math>G</math>, then <math>H \cap K</math> is 2-subnormal in <math>K</math>. {{proofat|[[2-subnormality satisfies transfer condition]]}}
+For more information on these operators: <toggledisplay>
-{{imagecondn}}
-If <math>f:G \to K</math> is a surjective homomorphism of groups, and <math>H</math> is 2-subnormal in <math>G</math>, then <math>f(H)</math> is 2-subnormal in <math>K</math>.
-{{not upper join-closed}}
-If <math>H \le G</math> and <math>K, L</math> are two intermediate subgroups containing <math>H</math>, it may happen that <math>H</math> is 2-subnormal in <math>K</math> as well as in <math>L</math>, but is not 2-subnormal in <math>\langle K, L \rangle</math>. {{proofat|[[2-subnormality is not upper join-closed]]}}
-==Effect of property operators==
 {{applyingoperatorgives|right transiter|right-transitively 2-subnormal subgroup}}
@@ Line 105: / Line 144: @@
 Any [[characteristic subgroup]] is left-transitively 2-subnormal, because the [[left transiter of normal is characteristic]].
+{{applyingoperatorgives|join-transiter|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.
+</toggledisplay>