Subnormal subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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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
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This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

Definition

Symbol-free definition

A subgroup of a group is termed subnormal if any of the following equivalent conditions holds:

There is a finite ascending chain of subgroups starting from the subgroup and going till the whole group, such that each is a normal subgroup of its successor. The smallest possible length of such a chain is termed the subnormal depth of the subnormal subgroup.
Consider the descending chain of subgroups defined as follows: each member is the normal closure of the original subgroup in its predecessor. This descending chain must reach the original subgroup within finitely many steps. The number of steps it takes is termed the subnormal depth.
The sequence of subgroups starting with the whole group, and where each is the commutator of its predecessor with the subgroup, gets inside the given subgroup after finitely many steps. The number of steps is takes is termed the subnormal depth.

The equivalence of the definitions thus must also show that the notions of subnormal depth in each case are the same.

A $k$ -subnormal subgroup is a subnormal subgroup with subnormal depth at most $k$ .

Definition with symbols

A subgroup $H$ is termed subnormal in a group $G$ if either of the following equivalent conditions holds:

There exists an ascending chain $H = H_{0}$ ≤ $H_{1}$ ... $H_{n} = G$ such that each $H_{i}$ is normal in $H_{i + 1}$ . The smallest possible $n$ for which such a chain exists is termed the subnormal depth of $H$ .
Consider the descending chain $G_{i}$ defined as follows: $G_{0} = G$ and $G_{i + 1}$ is the normal closure of $H$ in $G_{i}$ . Then, there exists an $n$ for which $G_{n} = H$ . The smallest such $n$ is termed the subnormal depth of $H$ .
Consider the sequence $K_{i}$ of subgroups of $G$ defined as follows: $K_{0} = G$ , and $K_{i + 1} = [H, K_{i}]$ , eventually enters inside $H$ . The number of steps taken is termed the subnormal depth of $H$ .

Equivalence of definitions

For full proof, refer: Equivalence of definitions of subnormal subgroup

Formalisms

In terms of the subordination operator

This property is obtained by applying the subordination operator to the property: normal subgroup
View other properties obtained by applying the subordination operator

The property of being a subnormal subgroup is obtained by applying the subordination operator to the group property of normality.

Relation with other properties

Stronger properties

Normal subgroup: The property of being normal is the same as that of being $1$ -subnormal.
Finitarily hypernormalized subgroup: For full proof, refer: Finitarily hypernormalized implies subnormal
2-subnormal subgroup
Permutable subgroup and Conjugate-permutable subgroup for the case of finite groups: For full proof, refer: Conjugate-permutable implies subnormal (finite groups)

Weaker properties

Conjunction with other properties

Conjunction with subnormal-to-normal subgroup gives normal subgroup
Conjunction with pronormal subgroup gives normal subgroup (this generalizes, in fact, to any property that is stronger than being subnormal-to-normal): For full proof, refer: Pronormal and subnormal implies normal

Related group properties

Subnormal intersection property is the property of being a group where an arbitrary intersection of subnormal subgroups is subnormal.
T-group is the property of being a group in which every subnormal subgroup is normal.

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this 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
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The property of being a subnormal subgroup is a transitive subgroup property. That is, any subnormal subgroup of a subnormal subgroup is 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

The property of being subnormal is trivially true, that is, the trivial subgroup is always subnormal.

The property of being subnormal is also identity-true, that is, every group is subnormal as a subgroup of itself.

Intersection-closedness

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

The property of being subnormal is a finite-intersection-closed subgroup property. That is, any finite intersection of subnormal subgroups is a subnormal subgroups.

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.
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The property of subnormality satisfies intermediate subgroup condition. That is, if $H$ is a subnormal subgroup of $G$ , and $K$ is an intermediate subgroup of $G$ containing $H$ , then $H$ is a subnormal subgroup of $K$ .

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
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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

Testing

The testing problem

Further information: subnormality testing problem

The property of being subnormal can be tested for permutation groups. In fact, it can be tested if we have an algorithm for normal closure-finding.

GAP command

This subgroup property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
View subgroup properties testable with built-in GAP command|View subgroup properties for which all subgroups can be listed with built-in GAP commands | View subgroup properties codable in GAP
Learn more about using GAP

The GAP syntax for determining whether a subgroup is subnormal is:

IsSubnormal (Group, Subgroup);

The GAP syntax for finding a subnormal series for a given subgroup is:

SubnormalSeries(Group, Subgroup);

The subnormal depth can be determined by the command:

Length(SubnormalSeries(Group, Subgroup)) - 1;

The -1 is necessary because the length of a series, in GAP, is the number of terms rather than the number of ascent/descent symbols.

Study of the notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20E15

Under the Mathematical subject classification, the study of a slight variant of this notion comes under the class: 20D35

The class 20D35 studies subnormal subgroups for finite groups. In the infinite case, 20E15 is more appropriate.

External links

Definition links