Subnormal subgroup

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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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Definition

Symbol-free definition

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

1. 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.
2. 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.
3. 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:

1. There exists an ascending chain $H = H_0 \le H_1 \le \dots \le H_n = G$ 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 $H$.
2. 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 $H$.
3. 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

Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

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.

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

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	(by definition)	If $H \le K \le G$, with $H$ subnormal in $K$ and $K$ subnormal in $G$, then $H$ is subnormal in $G$.
trim subgroup property	Yes	(by definition)	The trivial subgroup and the whole group are subnormal in the group.
finite-intersection-closed subgroup property	Yes	Subnormality is finite-intersection-closed	If $H_1, H_2$ are subnormal subgroups of $G$, then $H_1 \cap H_2$ is also subnormal in $G$.
finite-relative-intersection-closed subgroup property	Yes	Subnormality is finite-relative-intersection-closed	In other words, if $H, K \le G$ 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 $G$.
intermediate subgroup condition	Yes	Subnormality satisfies intermediate subgroup condition	If $H \le K \le G$ with $H$ subnormal in $G$, then $H$ is subnormal in $K$.
finite-join-closed subgroup property	No	Subnormality is not finite-join-closed	It is possible to have subnormal subgroups $H,K$ of a group $G$ such that $\langle H, K \rangle$ is not subnormal in $G$.
normalizing join-closed subgroup property	Yes	Subnormality is normalizing join-closed	If $H, K \le G$ are both subnormal subgroups and $K \le N_G(H)$, then $HK$ is also a subnormal subgroup.
permuting join-closed subgroup property	Yes	Subnormality is permuting join-closed	If $H, K \le G$ are both subnormal subgroups and $HK = KH$, then $HK$ is also a subnormal subgroup.
transfer condition	Yes	Subnormality satisfies transfer condition	If $H,K \le G$ with $H$ subnormal, then $H \cap K$ is subnormal in $K$.
centralizer-closed subgroup property	No	Subnormality is not centralizer-closed	It is possible to have $H \le G$ subnormal such that $C_G(H)$ is not subnormal in $G$.
commutator-closed subgroup property	No	Subnormality is not commutator-closed	It is possible to have subnormal subgroups $H, K \le G$ such that the commutator $[H,K]$ is not subnormal in $G$.
finite-upper join-closed subgroup property	No	Subnormality is not finite-upper join-closed	It is possible to have $H \le G$ with intermediate subgroups $K_1, K_2$ such that $H$ is subnormal in $K_1$ and in $K_2$ but not in $\langle K_1, K_2 \rangle$.
conditionally lattice-determined subgroup property	No	No subgroup property between normal Sylow and subnormal or between Sylow retract and retract is conditionally lattice-determined	It is possible to have a group $G$, an automorphism $\varphi$ of the lattice of subgroups, and a subnormal subgroup $H$ of $G$ such that $\varphi(H)$ is not subnormal.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Normal subgroup	subnormal of depth 1	(obvious)	subnormal not implies normal (also see normality is not transitive, there exist subgroups of arbitrarily large subnormal depth)	2-hypernormalized subgroup, 2-subnormal subgroup, 3-subnormal subgroup, 4-subnormal subgroup, Asymptotically fixed-depth join-transitively subnormal subgroup, Central factor of normal subgroup, Direct factor of normal subgroup, Finitarily hypernormalized subgroup, Finite-conjugate-join-closed subnormal subgroup, Intermediately join-transitively subnormal subgroup, Join of finitely many 2-subnormal subgroups, Join-transitively 2-subnormal subgroup, Join-transitively subnormal subgroup, Linear-bound join-transitively subnormal subgroup, Modular 2-subnormal subgroup, Modular subnormal subgroup, Normal subgroup of characteristic subgroup, Permutable 2-subnormal subgroup, Permutable subnormal subgroup, Subnormal-permutable subnormal subgroup|FULL LIST, MORE INFO
Finitarily hypernormalized subgroup	taking the normalizer finitely many times yields the whole group		follows from abnormal normalizer and 2-subnormal not implies normal	|FULL LIST, MORE INFO
2-subnormal subgroup	normal subgroup of normal subgroup	(obvious)	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	normal subgroup of normal subgroup of normal subgroup	(obvious)	there exist subgroups of arbitrarily large subnormal depth	|FULL LIST, MORE INFO
4-subnormal subgroup	normal of normal of normal of normal	(obvious)	there exist subgroups of arbitrarily large subnormal depth	|FULL LIST, MORE INFO
Join-transitively subnormal subgroup	join with any subnormal subgroup is subnormal	because the trivial subgroup is subnormal	subnormality is not finite-join-closed	Finite-conjugate-join-closed subnormal subgroup|FULL LIST, MORE INFO
Left-transitively fixed-depth subnormal subgroup			subnormal not implies left-transitively fixed-depth subnormal	|FULL LIST, MORE INFO
Right-transitively fixed-depth subnormal subgroup			subnormal not implies right-transitively fixed-depth subnormal	|FULL LIST, MORE INFO
Linear-bound join-transitively subnormal subgroup	join with a $k$-subnormal subgroup is subnormal of depth bounded by a linear function of $k$
Polynomial-bound join-transitively subnormal subgroup	join with a $k$-subnormal subgroup is subnormal of depth bounded by a polynomial function of $k$
Subnormal subgroup of finite group
Permutable subgroup of finite group		permutable implies subnormal in finite
Conjugate-permutable subgroup of finite group		conjugate-permutable implies subnormal in finite

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Ascendant subgroup	ascending (possibly transfinite) chain from subgroup to whole group, each normal in successor	subnormal implies ascendant	ascendant not implies subnormal	|FULL LIST, MORE INFO
Descendant subgroup		subnormal implies descendant	descendant not implies subnormal	Intersection of subnormal subgroups|FULL LIST, MORE INFO
Serial subgroup		(via ascendant, descendant)	(via ascendant, descendant)	Ascendant subgroup|FULL LIST, MORE INFO
Subpronormal subgroup				|FULL LIST, MORE INFO
Locally subnormal subgroup	subnormal in its join with any finitely generated subgroup			|FULL LIST, MORE INFO
Almost subnormal subgroup	ascending chain from subgroup to whole group with each member either normal or of finite index in successor			|FULL LIST, MORE INFO
Join of finitely many subnormal subgroups	join of finitely many subnormal subgroups
Join of subnormal subgroups	join of subnormal subgroups
Upper join of subnormal subgroups	the whole group is a join of intermediate subgroups in each of which the given subgroup is subnormal.

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.

Effect of property operators

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Operator	Meaning	Result of application	Proof
left transiter	Whenever the whole group is subnormal in some bigger group, so is the subgroup.	subnormal subgroup	subnormality is a t.i. subgroup property, so equals its own left transiter.
right transiter	Any subnormal subgroup of the subgroup is subnormal in the whole group.	subnormal subgroup	subnormality is a t.i. subgroup property, so equals its own right transiter.
subordination operator	There is a finite series of subgroups from the subgroup to the whole group with each subnormal in the next.	subnormal subgroup	subnormality is a t.i. subgroup property, so equals its own subordination.
join-transiter	The join with any subnormal subgroup is subnormal.	join-transitively subnormal subgroup	(by definition)
finite-join-closure operator	The join in the whole group of finitely many subnormal subgroups.	join of finitely many subnormal subgroups	(by definition)
join-closure operator	The join in the whole group of subnormal subgroups.	join of subnormal subgroups	(by definition)
intersection-closure operator	The intersection in the whole group of subnormal subgroups	intersection of subnormal subgroups	(by definition)

More on these property operators: [SHOW MORE]

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 $H \le G$ is termed join-transitively subnormal if $\langle H, K \rangle$ is subnormal for any subnormal subgroup $K$. Any normal subgroup, 2-subnormal subgroup or permutable subnormal subgroup is join-transitively subnormal.

The finite-join-closure operator

Applying the finite-join-closure operator to this property gives: join of finitely many subnormal subgroups

The join-closure operator

Applying the join-closure operator to this property gives: join of subnormal subgroups

The intersection-closure operator

Applying the intersection-closure operator to this property gives: intersection of subnormal subgroups

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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Definition links

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• Mathworld page (stub-length)