Normal subgroup
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Definition
QUICK PHRASES: invariant under inner automorphisms, self-conjugate subgroup, same left and right cosets, kernel of a homomorphism, subgroup that is a union of conjugacy classes
Equivalent definitions in tabular format
Note that each of these definitions (except the first one, as noted) assumes that we already have a group and a subgroup. To prove normality using any of these definitions, we first need to check that we actually have a subgroup.
No. | Shorthand | A subgroup of a group is normal in it if... | A subgroup of a group is normal in if ... | Applications to... | Additional comments |
---|---|---|---|---|---|
1 | Homomorphism kernel | it is the kernel of a homomorphism from the group. | there is a homomorphism from to a group such that the kernel of is precisely . In other words, is the identity element of if and only if . | proving normality | In this case, we do not need to separately check that is a subgroup since the kernel of a homomorphism is automatically a subgroup. |
2 | Inner automorphism invariance | it is invariant under all inner automorphisms. | for all , . More explicitly, for all , we have . | proving normality | Thus, normality is the invariance property with respect to the property of an automorphism being inner. This definition also motivates the term invariant subgroup for normal subgroup (which was used earlier). |
3 | Equals conjugates | it equals each of its conjugates in the whole group. | for all in , . | This definition also motivates the term self-conjugate subgroup for normal subgroup (which was used earlier). | |
4 | Left/right cosets equal | its left cosets are the same as its right cosets (that is, it commutes with every element of the group). | for all in , . | proving normality | When we say , we only mean equality as sets. It is not necessary that for . That stronger condition defines central subgroup. |
5 | Union of conjugacy classes | it is a union of conjugacy classes. | is a union of conjugacy classes in | ||
6 | Commutator inside | it contains its commutator with the whole group. | the commutator (which coincides with the commutator ) is contained in . | proving normality | |
7 | Conjugates of generating set inside | given a generating set for the whole group and a generating set for the subgroup, every conjugate of an element in the latter by an element in the former, as well as by its inverse, is in the subgroup. | given a generating set for and for , we have and for all . | normality testing problem | For finite groups, we need only check conjugates by elements in the generating set and not by their inverses. |
For more definitions, see nonstandard definitions of normal subgroup.
This definition is presented using a tabular format. |View all pages with definitions in tabular format
Notation and terminology
For a subgroup of a group , we denote the normality of in by or ^{Notations}. In words, we say that is normal in or a normal subgroup of .
Equivalence of definitions
Pair of definitions | Explanation of equivalence | More related information |
---|---|---|
(1) and (2) | Normal subgroup equals kernel of homomorphism | first isomorphism theorem |
(2) and (3) | Follows from the more general fact that restriction of automorphism to subgroup invariant under it and its inverse is automorphism, combined with the fact that the inverse of an inner automorphism is also an inner automorphism (in fact, the inverse of conjugation by is conjugation by ) | group acts as automorphisms by conjugation |
(3) and (4) | A direct manipulation of equations involving elements and subsets. For full proof, refer: equivalence of conjugacy and coset definitions of normality. | manipulating equations in groups |
(2) (or (3)) and (5) | A straightforward unraveling of the meaning of conjugacy class | |
(2) (or (3)) and (6) | A straightforward unraveling of the meaning of commutator, along with a little bit of manipulation. For full proof, refer: equivalence of conjugacy and commutator definitions of normality | manipulating equations in groups |
Copyable LaTeX
The following is LaTeX for a quick definition of normality.
A subgroup $H$ of a group $G$ is termed a {\em normal subgroup} if $ghg^{-1} \in H$ for all $g \in G$ and $h \in H$.
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
VIEW: Definitions built on this | Facts about this: (facts closely related to Normal subgroup, all facts related to Normal subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |[SHOW MORE]
This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties
View a list of pivotal subgroup properties | View a complete list of subgroup properties[SHOW MORE]
Importance
The notion of normal subgroup is important because of two main reasons:
- Normal subgroups are precisely the kernels of homomorphisms
- Normal subgroups are precisely the subgroups invariant under inner automorphisms, and for a group action, the only relevant automorphisms of the acting group that correspond to symmetries of the set being acted upon, are inner automorphisms.
Further information: Ubiquity of normality
Examples
VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions
Extreme examples
- The trivial subgroup is always normal. Further information: Trivial subgroup is normal
- Every group is normal as a subgroup of itself. Further information: Every group is normal in itself
Examples
- High occurrence example: In an abelian group, every subgroup is normal (there are non-abelian groups, such as the quaternion group, where every subgroup is normal. Groups in which every subgroup is normal are called Dedekind groups, and the non-abelian ones are called Hamiltonian groups). Further information: abelian implies every subgroup is normal
- If is an internal direct product of subgroups and , both and are normal in . Further information: direct factor implies normal
- Every subgroup-defining function yields a normal subgroup (in fact, it yields a characteristic subgroup). For instance, the center, derived subgroup and Frattini subgroup in any group are normal. Further information: subgroup-defining function value is characteristic, characteristic implies normal
Non-examples
Here are some examples of non-normal subgroups:
- In the symmetric group on three letters, the subgroup S2 in S3, i.e., the two-element subgroup generated by a transposition, is not normal (in fact, there are three such subgroups and they're all conjugate). Further information: S2 is not normal in S3
- More generally, in any dihedral group of degree at least , the two-element subgroup generated by a reflection is not normal. Further information: Two-element subgroup generated by reflection is not normal in dihedral group
- Low occurrence example: In a simple group, no proper nontrivial subgroup is normal. Thus, any proper nontrivial subgroup of a simple group gives a counterexample. The smallest simple non-Abelian group is the alternating group on five letters.
Subgroups satisfying the property
Here are examples of subgroups that satisfy the property of being normal:
Here are some examples of subgroups in basic/important groups satisfying the property:
Group part | Subgroup part | Quotient part | |
---|---|---|---|
A3 in S3 | Symmetric group:S3 | Cyclic group:Z3 | Cyclic group:Z2 |
Z2 in V4 | Klein four-group | Cyclic group:Z2 | Cyclic group:Z2 |
Here are some examples of subgroups in relatively less basic/important groups satisfying the property:
Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:
Subgroups dissatisfying the property
Here are examples of subgroups that do not satisfy the property of being normal.
Here are some examples of subgroups in basic/important groups not satisfying the property:
Group part | Subgroup part | Quotient part | |
---|---|---|---|
S2 in S3 | Symmetric group:S3 | Cyclic group:Z2 |
Here are some some examples of subgroups in relatively less basic/important groups not satisfying the property:
Here are some examples of subgroups in even more complicated/less basic groups not satisfying the property:
Group part | Subgroup part | Quotient part | |
---|---|---|---|
2-Sylow subgroup of general linear group:GL(2,3) | General linear group:GL(2,3) | Semidihedral group:SD16 | |
Non-normal subgroups of M16 | M16 | Cyclic group:Z2 |
Facts
Isomorphism theorems
Theorem name/number | Statement |
---|---|
First isomorphism theorem | If is a surjective homomorphism, and is the kernel of , then is a normal subgroup, and if is the quotient map, then there is a unique isomorphism such that . |
Second isomorphism theorem (diamond isomorphism theorem) | If are subgroups such that is contained in the normalizer of , then is normal in and . |
Third isomorphism theorem | If are groups with both normal in , then is normal in and . |
Fourth isomorphism theorem (lattice isomorphism theorem) | If is normal in , then there is a bijective correspondence between subgroups of and subgroups of containing , satisfying many nice conditions. |
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 |
---|---|---|---|
abelian-tautological subgroup property | Yes | abelian implies every subgroup is normal | If and is abelian, then is normal in . |
transitive subgroup property | No | Normality is not transitive | We can have such that is normal in and is normal in but is not normal in . |
trim subgroup property | Yes | Every group is normal in itself, trivial subgroup is normal | trivial subgroup and whole group are both normal |
strongly intersection-closed subgroup property | Yes | Normality is strongly intersection-closed | If are all normal subgroups of a group , then the intersection of subgroups is also a normal subgroup of . |
strongly join-closed subgroup property | Yes | Normality is strongly join-closed | If are all normal subgroups of , then the join of subgroups is also a normal subgroup of . |
quotient-transitive subgroup property | Yes | Normality is quotient-transitive | If are groups such that is normal in and is normal in , then is normal in |
intermediate subgroup condition | Yes | Normality satisfies intermediate subgroup condition | If are groups such that is normal in , then is normal in . |
transfer condition | Yes | Normality satisfies transfer condition | If are groups such that is normal in , then we must have that normal in |
image condition | Yes | Normality satisfies image condition | If is a normal subgroup of and is a surjective homomorphism of groups, then normal in |
inverse image condition | Yes | Normality satisfies inverse image condition | If is a normal subgroup of and is a homomorphism of groups, then is normal in . |
upper join-closed subgroup property | Yes | Normality is upper join-closed | If , and are all subgroups of containing such that normal in each , then we must have that is normal in the join of subgroups . |
commutator-closed subgroup property | Yes | Normality is commutator-closed | If are both normal subgroups of a group , then the commutator normal in |
centralizer-closed subgroup property | Yes | Normality is centralizer-closed | If is a normal subgroup of , the centralizer is also normal in . |
direct product-closed subgroup property | Yes | Normality is direct product-closed | If is an indexing set, are groups, and are such that each is normal in the corresponding , then in the external direct product of the s, the subgroup given by the external direct product of the s is a normal subgroup. This also implies that normality is a finite direct power-closed subgroup property. |
strongly UL-intersection-closed subgroup property | Yes | Normality is strongly UL-intersection-closed | If is an indexing set, is a group, and we have groups for each , such that is normal in , then is a normal subgroup of . |
Arguesian subgroup property | Yes | Normality is Arguesian | The collection of normal subgroups of a group form an Arguesian lattice. |
lower central series condition | Yes | Normality satisfies lower central series condition | Suppose is a normal subgroup of a group . Then, each lower central series member is a normal subgroup onside the corresponding lower central series member . |
partition difference condition | Yes | Normality satisfies partition difference condition | Suppose is a subgroup of a group , and has a partition as a union of . If all except possibly one of the s are normal subgroups of , then all the s are normal subgroups of . |
Relation with other properties
This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied
Some of these can be found at:
To get a broad overview, check out the survey articles:
- Varying normality
- Subnormal-to-normal and normal-to-characteristic
- Between normal and characteristic and beyond
- Between normal and subnormal and beyond
- Contrasting subnormality of various depths
Stronger properties
The most important stronger property is characteristic subgroup. See the table below for many stronger properties and the way they're related:
Other less important properties that are stronger than normality:
For a complete list of subgroup properties stronger than Normal subgroup, click here
STRONGER PROPERTIES SATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | inverse image condition | image condition | centralizer-closed |
STRONGER PROPERTIES DISSATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | inverse image condition | image condition | centralizer-closed |
Conjunction with other properties
Important conjunctions of normality with other subgroup properties are in the table below:
View a complete list of conjunctions of normality with subgroup properties
We are often also interested in the conjunction of normality with group properties. By this, we mean the subgroup property of being normal as a subgroup and having the given group property as an abstract group. Examples are in the table below:
View a complete list of conjunctions of normality with group properties
In some cases, we are interested in studying normal subgroups with the big group constrained to satisfy some group property. For instance:
Weaker properties
Other less important properties that are weaker than normality:
For a complete list of subgroup properties weaker than Normal subgroup, click here
WEAKER PROPERTIES SATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | inverse image condition | image condition | centralizer-closed |
WEAKER PROPERTIES DISSATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | inverse image condition | image condition | centralizer-closed |
Related operators
There are three important subgroup operators related to normality:
Operator | What it does | How normal subgroups are related to it |
---|---|---|
Normal core | This takes a subgroup and outputs the largest normal subgroup inside it, which is also the intersection of all its conjugate subgroups | The operator is idempotent (doing it twice is the same as doing it once) and normal subgroups are precisely the subgroups that are invariant under it, and hence also precisely the subgroups that can arise from it. |
Normal closure | This takes a subgroup and outputs the smallest normal subgroup containing it, which is also the join of all its conjugate subgroups | The operator is idempotent (doing it twice is the same as doing it once) and normal subgroups are precisely the subgroups that are invariant under it, and hence also precisely the subgroups that can arise from it. |
Normalizer | This takes a subgroup and outputs the largest subgroup within which it is normal | A subgroup is normal if and only if its normalizer is the whole group |
Other operators involve composing these in different ways, for instance:
Closely related to normal closure is the normal subgroup generated by a subset, which is defined as the smallest normal subgroup containing the subset, and is the normal closure of the subgroup generated by the subset.
Analogues in other algebraic structures
Algebraic structure | Analogue of normal subgroup in that structure | Definition | Nature of analogy with normal subgroup |
---|---|---|---|
Lie ring | Ideal of a Lie ring | A subring of a Lie ring whose Lie bracket with any element of the Lie ring is in the subring. | Precisely the kernels of homomorphisms, play analogous roles in isomorphism theorems. Also, precisely the subrings invariant under inner derivations. Also, the Lazard correspondence maps ideals of the Lazard Lie ring to normal subgroups of the group. |
ideal-determined variety of algebras (universal algebra) | ideal in that variety | An ideal-determined variety of algebras with zero is a variety where every ideal occurs as the inverse image of zero under some homomorphism and this completely determines the fibers of the homomorphism. Also, the various isomorphism theorems hold with suitable modifications. | variety of groups is ideal-determined, with the ideals being the normal subgroups. |
variety of algebras (universal algebra) | I-automorphism-invariant subalgebra | invariant under I-automorphisms, which are the automorphism described by formulas that universally give automorphisms. | inner automorphisms are I-automorphisms in the variety of groups, so the corresponding invariant subalgebras are subgroups. |
loop | normal subloop | commutes with every element, associates with every pair of elements | when the loop is a group, then normal subloop = normal subgroup. Also, normal subloops are kernels of homomorphisms and the isomorphism theorems hold. |
hypergroup | normal subhypergroup | ||
semigroup | left-normal subsemigroup, right-normal subsemigroup |
Effect of property operators
Operator | Meaning | Result of application | Proof |
---|---|---|---|
left transiter | if big group is normal in a bigger group, so is subgroup | characteristic subgroup | left transiter of normal is characteristic |
right transiter | every normal subgroup of subgroup is normal in whole group | transitively normal subgroup | by definition |
subordination operator | normal subgroup of normal subgroup of ... of normal subgroup | subnormal subgroup | by definition |
hereditarily operator | every subgroup of it is normal | hereditarily normal subgroup | by definition |
upward-closure operator | every subgroup containing it is normal | upward-closed normal subgroup | by definition |
maximal proper operator | proper normal subgroup contained in no other proper normal subgroup | maximal normal subgroup | by definition |
minimal operator | nontrivial normal subgroup containing no other nontrivial normal subgroup | minimal normal subgroup | by definition |
Formalisms
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)
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
The subgroup property of normality can be expressed in first-order language as follows: is normal in if and only if:
This is in fact a universally quantified expression of Fraisse rank 1.
Function restriction expression
This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
Function restriction expression | is a normal subgroup of if ... | This means that normality is ... | Additional comments |
---|---|---|---|
inner automorphism function | every inner automorphism of sends every element of to within | the invariance property for inner automorphisms | |
inner automorphism endomorphism | every inner automorphism of restricts to an endomorphism of | the endo-invariance property for inner automorphisms; i.e., it is the invariance property for inner automorphism, which is a property stronger than the property of being an endomorphism | |
inner automorphism automorphism | every inner automorphism of restricts to an automorphism of | the auto-invariance property for inner automorphisms; i.e., it is the invariance property for inner automorphism, which is a group-closed property of automorphisms | inner automorphism to automorphism is right tight for normality |
Relation implication expression
This subgroup property is a relation implication-expressible subgroup property: it can be defined and viewed using a relation implication expression
View other relation implication-expressible subgroup properties
Normality can be expressed in terms of the relation implication formalism as the relation implication operator with the left side being conjugate subgroups and the right side being equal subgroups:
Conjugate Equal
In other words, a subgroup is normal if any subgroup related to it by being conjugate is in fact equal to it.
Variety formalism
This subgroup property can be described in the language of universal algebra, viewing groups as a variety of algebras
View other such subgroup properties
There are two somewhat different ways of expressing the notion of normality in the language of varieties:
- In the variety of groups, the normal subgroups are precisely the subalgebras invariant under all the I-automorphisms. An I-automorphism is an automorphism that can be expressed using a formula guaranteed to give an automorphism. This definition of normal subgroup follows from the fact that for groups, inner automorphisms are precisely the I-automorphisms.
- Treating the variety of groups as a variety of algebras with zero, the normal subgroups are precisely the ideals.
Testing
The testing problem
Further information: Normality testing problem
Given generating sets for a group and a subgroup, the problem of determining whether the subgroup is normal in the group reduces to the problem of testing whether the conjugate of any generator of the subgroup by any generator of the group is inside the subgroup. Thus, it reduces to the membership problem for the subgroup.
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:IsNormal
The GAP command for listing all subgroups with this property is:NormalSubgroups
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 testing whether a subgroup is normal in a group is:
IsNormal (group, subgroup);
where subgroup and group may be defined on the spot in terms of generators (described as permutations) or may refer to things previously defined.
GAP can also be used to list all normal subgroups of a given group, using the command:
NormalSubgroups(group);
History
[SHOW MORE]References
Textbook references
Advanced undergraduate/beginning graduate algebra texts that include group theory:
Book | Page number | Chapter and section | Contextual information | View |
---|---|---|---|---|
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347^{More info} | 82 | formal definition,, and Theorem 6 giving equivalent formulations. Also, Page 80 (first use). | ||
Topics in Algebra by I. N. Herstein^{More info} | 50 | Section 2.6 | formal definition | |
Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189^{More info} | 41 | Section 1.5 | definition introduced through proposition | Google Books |
Algebra by Serge Lang, ISBN 038795385X^{More info} | 14 | definition in paragraph | Google Books | |
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632^{More info} | 52 | Point (4.8) | formal definition, followed by equivalent definition-cum-proposition in (4.9) | |
Basic Algebra: Groups, Rings, and Fields by Paul Moritz Cohn, ISBN 1852335874, 13-digit ISBN 978-1852335878^{More info} | 27 | Section 2.1 | definition in paragraph. This book uses a right action convention for functions. | Google Books |
Schaum's outline of group theory by Benjamín Baumslag and Bruce Chandler, ISBN 0070041245, 13-digit ISBN 978-0070041240^{More info} | 111 | formal definition after some motivating discussion | Google Books |
Graduate texts on group theory:
Book | Page number | Chapter and section | Contextual information | View |
---|---|---|---|---|
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261^{More info} | 6 | definition in paragraph | Google Books | |
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info} | 15 | Proposition 1.3.15 | definition introduced through proposition | Google Books |
An introduction to the theory of groups by Joseph J. Rotman, ISBN 0387942858, 13-digit ISBN 978-0387942858^{More info} | 30 | formal definition | Google Books |
Online lecture notes
- J.S. Milne's course notes, Section 1.7, Page 15 (both the A4 and the letter versions)
- Lecture notes for Benedict Gross's lecture 4, Page 2-3 and more. You can also access the video lectures and other material for the course through the course main page.
- Lecture notes on basic group theory by James Miller, part of the solitaryroad.com website