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 of a group if 2-subnormal in 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 of such that is a normal subgroup of and is a normal subgroup of .
|
2 |
Normal in closure |
the subgroup is normal in its normal closure in the whole group. |
is a normal subgroup of its normal closure in .
|
3 |
Normal closure in normalizer |
the normal closure of the subgroup is contained in the normalizer of the subgroup. |
the normal closure is contained in the normalizer .
|
4 |
Every conjugate in normalizer |
every conjugate of the subgroup is contained in its normalizer, i.e., every conjugate normalizes it. |
for every , .
|
5 |
In normal core of normalizer |
the subgroup is contained in its normal core of normalizer: the normal core of its normalizer. |
is contained in the normal core of in .
|
6 |
Subnormal of depth 2 |
it is a subnormal subgroup whose subnormal depth (also called subnormal defect) is at most . |
is subnormal in with subnormal depth at most .
|
7 |
contains second commutator |
it contains its second commutator subgroup with the whole group |
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
A 2-subnormal subgroup
has a unique fastest ascending subnormal series
, where
is the normal core of
. It also has a unique fastest descending subnormal series
, where
is the normal closure of
in
. 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
is 2-subnormal in a group
if it satisfies the following first-order sentence:
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 , with 2-subnormal in , 2-subnormal in , but not 2-subnormal in .
|
trim subgroup property |
Yes |
Every group is normal in itself, trivial subgroup is normal |
For any group , 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 |
all 2-subnormal subgroups of , then so is .
|
finite-join-closed subgroup property |
No |
2-subnormality is not finite-join-closed |
Can have subgroups , both 2-subnormal in , such that is not 2-subnormal.
|
conjugate-join-closed subgroup property |
Yes |
2-subnormality is conjugate-join-closed |
A join of subgroups of , with all 2-subnormal in and all conjugate to each other, is also 2-subnormal.
|
intermediate subgroup condition |
Yes |
2-subnormality satisfies intermediate subgroup condition |
If , with 2-subnormal in , then is 2-subnormal in .
|
transfer condition |
Yes |
2-subnormality satisfies transfer condition |
If , with 2-subnormal in , then is 2-subnormal in .
|
image condition |
Yes |
2-subnormality satisfies image condition |
If 2-subnormal in , surjective homomorphism, then is 2-subnormal in .
|
inverse image condition |
Yes |
2-subnormality satisfies inverse image condition |
If homomorphism, 2-subnormal in , then 2-subnormal in .
|
upper join-closed subgroup property |
No |
2-subnormality is not upper join-closed |
Can have with 2-subnormal in both and but not in .
|
For more details of these metaproperties:
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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
is a 2-subnormal subgroup of
, then
is also 2-subnormal in any intermediate subgroup
of
. 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
is a 2-subnormal subgroup of
and
is any subgroup of
, then
is 2-subnormal in
. 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
is a surjective homomorphism of groups, and
is 2-subnormal in
, then
is 2-subnormal in
.
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
and
are two intermediate subgroups containing
, it may happen that
is 2-subnormal in
as well as in
, but is not 2-subnormal in
. For full proof, refer: 2-subnormality is not upper join-closed
Effect of property operators
For more information on these operators:
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