Subnormal subgroup

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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 $H = H_{0} \leq H_{1} \dots H_{n} = G$ 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 $K_{i + 1} = [H, K_{i}]$ (the commutator), This sequence of subgroups eventually enters inside $H$ . The number of steps taken is termed the subnormal depth of $H$ .

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

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: For proof of the implication, refer Normal implies subnormal and for proof of its strictness (i.e. the reverse implication being false) refer Subnormal not implies normal.
Finitarily hypernormalized subgroup: For full proof, refer: Finitarily hypernormalized implies subnormal
2-subnormal subgroup
Join-transitively 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

Ascendant subgroup: For proof of the implication, refer Subnormal implies ascendant and for proof of its strictness (i.e. the reverse implication being false) refer Ascendant not implies subnormal.
Descendant subgroup: For proof of the implication, refer Subnormal implies descendant and for proof of its strictness (i.e. the reverse implication being false) refer Descendant not implies subnormal.
Serial subgroup
Subpronormal subgroup

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

In fact, there are a large number of subgroup properties whose conjunction with subnormality gives normality. Further information: subnormal-to-normal and normal-to-characteristic

Conjunction with group properties:

Abelian subnormal subgroup is a subnormal subgroup that is also an abelian group. Also related:
- Nilpotent subnormal subgroup is a subnormal subgroup that is also a nilpotent group.
- Cyclic subnormal subgroup is a subnormal subgroup that is also a cyclic group.
- Solvable subnormal subgroup is a subnormal subgroup that is also a solvable group.
Finite subnormal subgroup is a subnormal subgroup that is also a finite group.
Perfect subnormal subgroup is a subnormal subgroup that is also a perfect group. Also related:
- Simple subnormal subgroup is a subnormal subgroup that is also a simple group.
- Component is a subnormal subgroup that is also a quasisimple group.

The property of being subnormal in particular kinds of groups is also of interest:

Subgroup of nilpotent group is a subnormal subgroup of a nilpotent group. (Note that nilpotent implies every subgroup is subnormal).
Subnormal subgroup of finite group is a subnormal subgroup of a finite group.
Subnormal subgroup of solvable group is a subnormal subgroup of a solvable group.

Related group properties

Group in which every subgroup is subnormal is a group in which every subgroup is subnormal. Nilpotent groups have this property: nilpotent implies every subgroup is subnormal.
Subnormal intersection property is the property of being a group where an arbitrary intersection of subnormal subgroups is subnormal.
Subnormal join property is the property of being a group where a join of finitely many subnormal subgroups is subnormal.
Generalized subnormal join property is the property of being a group where an arbitrary join of subnormal subgroups is subnormal.
T-group is the property of being a group in which every subnormal subgroup is normal.

Facts

There exist subgroups of arbitrarily large subnormal depth: For any positive integer $k$ , there exists a group $G$ and a subgroup $H$ of $G$ such that the subnormal depth of $H$ in $G$ is precisely $k$ ..

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
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

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. In other words, any finite intersection of subnormal subgroups is a subnormal subgroups.

More generally, an arbitrary intersection of $k$ -subnormal subgroups is $k$ -subnormal. The reason why an infinite intersection of subnormal subgroups need not be subnormal is that the subnormal depth of these subgroups need not be bounded. Any arbitrary intersection of subnormal subgroups is a descendant subgroup. For full proof, refer: Subnormality of fixed depth is strongly intersection-closed, Subnormality is not intersection-closed, Subnormality is finite-intersection-closed, Descendant not implies subnormal

Relative-intersection-closedness

This subgroup property is finite-relative-intersection-closed.
View a complete list of finite-relative-intersection-closed subgroup properties

The property of being subnormal is a finite-relative-intersection-closed subgroup property. In other words, if $H, K \leq G$ are subgroups such that $H$ is $h$ -subnormal in $G$ and $K$ is $k$ -subnormal in some subgroup $L$ of $G$ containing both $H$ and $K$ , then $H \cap K$ is $(h + k)$ -subnormal in $G$ .

For full proof, refer: Subnormality is finite-relative-intersection-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

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 $K$ . Further information: Subnormality satisfies intermediate subgroup condition

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

A join of two subnormal subgroups need not be subnormal. A group where a join of two subnormal subgroups is always subnormal is termed a group satisfying subnormal join property, while a group where a join of arbitrarily many subnormal subgroups is subnormal is termed a group satisfying generalized subnormal join property. All finite groups have these properties; in a finite group, a join of subnormal subgroups is always subnormal. For full proof, refer: Subnormality is not join-closed, subnormality is not finite-join-closed

Also refer the sections on normalizing joins, permuting joins, and join-transiters.

Normalizing joins

This subgroup property is normalizing join-closed: the join of two subgroups with the property, one of which normalizes the other, also has the property.
View other normalizing join-closed subgroup properties

If $H, K \leq G$ are subnormal and $K \leq N_{G} (H)$ , then the join of $H$ and $K$ is subnormal; in fact, its subnormal depth is bounded by the product of subnormal depths of $H$ and $K$ . For full proof, refer: Subnormality is normalizing join-closed

Permuting joins

This subgroup property is permuting join-closed: the join of two permuting subgroups, both of which have the property, also has the property.
View other permuting join-closed subgroup properties

If $H, K \leq G$ are subnormal and they permute, i.e., $H K = K H$ , then the product $H K$ (which is also the join) is also subnormal. Further, its subnormal depth is bounded by a function of the subnormal depths of $H$ and $K$ . For full proof, refer: Subnormality is permuting join-closed

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 subnormal subgroup of $G$ , then for any subgroup $K \leq G$ , the intersection $H \cap K$ is a subnormal subgroup of $K$ . Further, the subnormal depth of $H \cap K$ in $K$ is bounded by the subnormal depth of $H$ in $G$ .

For full proof, refer: Subnormality satisfies transfer condition

Commutator-closedness

NO: This subgroup property is not commutator-closed: the commutator of two subgroups each with the property, need not have the property.
View other subgroup properties that are not commutator-closed

We can have two subnormal subgroups $H, K$ of a group $G$ such that $[H, K]$ is not a subnormal subgroup.

Effect of property operators

The left transiter

Applying the left transiter to this property gives: subnormal subgroup

Since subnormality is a t.i. subgroup property, it equals its own left transiter. However, there is a stronger property of being a left-transitively fixed-depth subnormal subgroup.

The right transiter

Applying the right transiter to this property gives: subnormal subgroup

Since subnormality is a t.i. subgroup property, it equals its own right transiter. However, there is a stronger property of being a right-transitively fixed-depth subnormal subgroup.

The join-transiter

Applying the join-transiter to this property gives: join-transitively subnormal subgroup

A subgroup $H \leq G$ is termed join-transitively subnormal if $⟨ H, K ⟩$ is subnormal for any subnormal subgroup $K$ . Any normal subgroup, 2-subnormal subgroup or permutable subnormal subgroup is join-transitively subnormal.

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).
The GAP command for testing this subgroup property is:IsSubnormal
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.

References

Textbook references

A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info}, Page 63 (definition introduced in the context of a more general definition)
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261^{More info}, Page 92-93 (introduces notion of subnormal series, does not explicitly talk of subnormal subgroup)
Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754^{More info}, Page 23 (formal definition)
Finite Groups by Daniel Gorenstein, ISBN 0821843427^{More info}, Page 13, Chapter 1, Exercise 5 (definition introduced in exercise)

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