Subnormal subgroup
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 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:
- 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 -subnormal subgroup is a subnormal subgroup with subnormal depth at most .
Definition with symbols
A subgroup is termed subnormal in a group if either of the following equivalent conditions holds:
- There exists an ascending chain such that each is normal in . The smallest possible for which such a chain exists is termed the subnormal depth of .
- Consider the descending chain defined as follows: and is the normal closure of in . Then, there exists an for which . The smallest such is termed the subnormal depth of .
- Consider the sequence of subgroups of defined as follows: , and (the commutator), This sequence of subgroups eventually enters inside . The number of steps taken is termed the subnormal depth of .
A -subnormal subgroup is a subnormal subgroup with subnormal depth at most .
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 , there exists a group and a subgroup of such that the subnormal depth of in is precisely ..
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 , with subnormal in and subnormal in , then is subnormal in . |
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 are subnormal subgroups of , then is also subnormal in . |
finite-relative-intersection-closed subgroup property | Yes | Subnormality is finite-relative-intersection-closed | In other words, if are subgroups such that is -subnormal in and is -subnormal in some subgroup of containing both and , then is -subnormal in . |
intermediate subgroup condition | Yes | Subnormality satisfies intermediate subgroup condition | If with subnormal in , then is subnormal in . |
finite-join-closed subgroup property | No | Subnormality is not finite-join-closed | It is possible to have subnormal subgroups of a group such that is not subnormal in . |
normalizing join-closed subgroup property | Yes | Subnormality is normalizing join-closed | If are both subnormal subgroups and , then is also a subnormal subgroup. |
permuting join-closed subgroup property | Yes | Subnormality is permuting join-closed | If are both subnormal subgroups and , then is also a subnormal subgroup. |
transfer condition | Yes | Subnormality satisfies transfer condition | If with subnormal, then is subnormal in . |
centralizer-closed subgroup property | No | Subnormality is not centralizer-closed | It is possible to have subnormal such that is not subnormal in . |
commutator-closed subgroup property | No | Subnormality is not commutator-closed | It is possible to have subnormal subgroups such that the commutator is not subnormal in . |
finite-upper join-closed subgroup property | No | Subnormality is not finite-upper join-closed | It is possible to have with intermediate subgroups such that is subnormal in and in but not in . |
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 , an automorphism of the lattice of subgroups, and a subnormal subgroup of such that is not subnormal. |
Relation with other properties
Stronger properties
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) |
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)
External links
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Definition links
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