From Groupprops
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 ...
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| 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.
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| 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 HG in G.
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| 3 | Normal closure in normalizer | the normal closure of the subgroup is contained in the normalizer of the subgroup. | the normal closure HG is contained in the normalizer NG(H).
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| 4 | Every conjugate in normalizer | every conjugate of the subgroup is contained in its normalizer, i.e., every conjugate normalizes it. | for every , .
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| 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 (NG(H))G of NG(H) in G.
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| 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.
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| 7 | contains second commutator | it contains its second commutator subgroup with the whole group | 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.[SHOW MORE]
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 H has a unique fastest ascending subnormal series
, where K is the normal core of NG(H). It also has a unique fastest descending subnormal series
, 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:
Examples
VIEW: |
VIEW: |
Relation with other properties
Stronger properties
Weaker properties
Metaproperties
| Metaproperty name | Satisfied? | Proof | Statement with symbols
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| transitive subgroup property | No | 2-subnormality is not transitive | There exist groups , with H 2-subnormal in K, K 2-subnormal in G, but H not 2-subnormal in G.
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| 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.
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| strongly intersection-closed subgroup property | Yes | Subnormality of fixed depth is strongly intersection-closed | all 2-subnormal subgroups of G, then so is .
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| finite-join-closed subgroup property | No | 2-subnormality is not finite-join-closed | Can have subgroups , both 2-subnormal in G, such that is not 2-subnormal.
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| conjugate-join-closed subgroup property | Yes | 2-subnormality is conjugate-join-closed | A join of subgroups of G, with all 2-subnormal in G and all conjugate to each other, is also 2-subnormal.
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| intermediate subgroup condition | Yes | 2-subnormality satisfies intermediate subgroup condition | If , with H 2-subnormal in G, then H is 2-subnormal in K.
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| transfer condition | Yes | 2-subnormality satisfies transfer condition | If , with H 2-subnormal in G, then is 2-subnormal in K.
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| image condition | Yes | 2-subnormality satisfies image condition | If H 2-subnormal in G, surjective homomorphism, then f(H) is 2-subnormal in K.
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| inverse image condition | Yes | 2-subnormality satisfies inverse image condition | If homomorphism, H 2-subnormal in G, then f − 1(H) 2-subnormal in K.
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| upper join-closed subgroup property | No | 2-subnormality is not upper join-closed | Can have with H 2-subnormal in both K and L but not in .
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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: |
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: |
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: |
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
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
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
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
. 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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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 + |