The Group Properties Wiki (pre-alpha)

TIP: Read more about how the definition in Groupprops is structured

ABOUT US: Read our purpose statement and learn what makes us special

ALSO CHECK OUT: Diffgeom: The Differential Geometry Wiki

2-subnormal subgroup

From Groupprops

Jump to: navigation, search

Contents

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 | Survey articles about 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
View a complete list of variations of normality OR read a survey article on varying normality

Definition

QUICK PHRASES: 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 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 H of a group G is termed 2-subnormal if the following equivalent conditions hold:

  • There is subgroup K such that H is a normal subgroup of K and K is a normal subgroup of G.
  • The normal closure of H is a normal subgroup of G.

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

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

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
View a complete list of transitive subgroup properties|View facts related to transitivity of subgroup properties

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

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 all trim subgroup properties OR view trivially true subgroup properties OR view 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.

Intersection-closedness

This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property
View a complete list of intersection-closed subgroup properties

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 an intersection 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

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

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
View all subgroup properties satisfying the intermediate subgroup condition|View facts related to the 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

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 a complete list of such properties

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

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 a complete list of subgroup properties satisfying the 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

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

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.

Personal tools