Normal subgroup

From Groupprops

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This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties
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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

Symbol-free definition

A subgroup of a group is said to be normal if it satisfies the following equivalent conditions:

(Homomorphism kernel definition): It is the kernel of a homomorphism from the group.
(Inner automorphisms definition): It is invariant under all inner automorphisms. 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).
(Equals conjugates definition): It equals each of its conjugates in the whole group. This definition also motivates the term self-conjugate subgroup for normal subgroup (which was used earlier).
(Cosets definition): Its left cosets are the same as its right cosets (that is, it commutes with every element of the group).
(Conjugacy class definition): It is a union of conjugacy classes.
(Commutator definition): It contains its commutator with the whole group.

Definition with symbols

A subgroup $N\!$ of a group $G\!$ is said to be normal in $G$ (in symbols, $N \triangleleft G$ or $G \triangleright N$ ^Notations) if the following equivalent conditions hold:

(Homomorphism kernel definition): There is a homomorphism $\varphi$ from $G$ to a group $K$ such that the kernel of $\varphi$ is precisely $N\!$ . In other words, $\varphi(x) \!$ is the identity element of $K$ if and only if $x \in N$ .
(Inner automorphisms definition): For all $g \in G$ , $gNg^{-1} \subseteq N$ . More explicitly, for all $g \in G, h \in N$ , we have $ghg^{-1} \in N$ .
(Equals conjuate definition): For all $g$ in $G$ , $g N g - 1 = N$ .
(Cosets definition): For all $g$ in $G$ , $g N = N g$ .
(Conjugacy classes definition): $N$ is a union of conjugacy classes.
(Commutator definition): The commutator $[N, G]$ (which coincides with the commutator $[G, N]$ ) is contained in $N$ .

Equivalence of definitions

For the equivalence between definitions (1) and (2), refer Normal subgroup equals kernel of homomorphism.
The equivalence between definitions (2) and (3) follows from a more general fact: 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 $g$ is conjugation by $g - 1$ ). Further information: group acts as automorphisms by conjugation
The equivalence between definitions (3) and (4) is a direct manipulation of equations involving elements and subsets. Further information: manipulating equations in groups
The equivalence of definitions (2) or (3) with definition (5) is a straightforward unraveling of the meaning of conjugacy class.
The equivalence of definitions (2) or (3) with definition (6) involves a straightforward unraveling of the meaning of commutator, along with a little bit of manipulation. Further information: manipulating equations in groups

Want more definitions? Check out nonstandard definitions of normal subgroup and historical definitions of normal subgroup. Many of these definitions are actually useful!

QUICK BITE: There's something special about groups that makes it true that the kernel of a homomorphism is actually a subgroup. Learn more at ideals are subalgebras in the variety of groups and characteristic subalgebras are ideals in the variety of groups

History

[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 |

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 $G$ is an internal direct product of subgroups $H$ and $K$ , both $H$ and $K$ are normal in $G$ . 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, commutator 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 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 $3$ , 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.

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: $N$ is normal in $G$ if and only if:

$\forall g \in G, h \in N: \ ghg^{-1} \in N$

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	$H$ is a normal subgroup of $G$ if ...	This means that normality is ...	Additional comments
inner automorphism $\to$ function	every inner automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for inner automorphisms
inner automorphism $\to$ endomorphism	every inner automorphism of $G$ restricts to an endomorphism of $H$	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 $\to$ automorphism	every inner automorphism of $G$ restricts to an automorphism of $H$	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

[SHOW MORE]

Function restriction expression	$H$ is a normal subgroup of $G$ if ...	This means that normality is ...	Additional comments
class-preserving automorphism $\to$ function	every class-preserving automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for class-preserving automorphisms
class-preserving automorphism $\to$ endomorphism	every class-preserving automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for class-preserving automorphisms; i.e., it is the invariance property for class-preserving automorphism, which is a property stronger than the property of being an endomorphism
class-preserving automorphism $\to$ automorphism	every class-preserving automorphism of $G$ restricts to an automorphism of $H$	the auto-invariance property for class-preserving automorphisms; i.e., it is the invariance property for class-preserving automorphism, which is a group-closed property of automorphisms	class-preserving automorphism to automorphism is right tight for normality
subgroup-conjugating automorphism $\to$ function	every subgroup-conjugating automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for subgroup-conjugating automorphisms
subgroup-conjugating automorphism $\to$ endomorphism	every subgroup-conjugating automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for subgroup-conjugating automorphisms; i.e., it is the invariance property for subgroup-conjugating automorphism, which is a property stronger than the property of being an endomorphism
subgroup-conjugating automorphism $\to$ automorphism	every subgroup-conjugating automorphism of $G$ restricts to an automorphism of $H$	the auto-invariance property for subgroup-conjugating automorphisms; i.e., it is the invariance property for subgroup-conjugating automorphism, which is a group-closed property of automorphisms
normal automorphism $\to$ function	every normal automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for normal automorphisms
normal automorphism $\to$ endomorphism	every normal automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for normal automorphisms; i.e., it is the invariance property for normal automorphism, which is a property stronger than the property of being an endomorphism
normal automorphism $\to$ automorphism	every normal automorphism of $G$ restricts to an automorphism of $H$	the auto-invariance property for normal automorphisms; i.e., it is the invariance property for normal automorphism, which is a group-closed property of automorphisms	normal automorphism to automorphism is left tight for normality (also right tight)
weakly normal automorphism $\to$ function	every weakly normal automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for weakly normal automorphisms
weakly normal automorphism $\to$ endomorphism	every weakly normal automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for weakly normal automorphisms; i.e., it is the invariance property for weakly normal automorphism, which is a property stronger than the property of being an endomorphism
monomial automorphism $\to$ function	every monomial automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for monomial automorphisms
monomial automorphism $\to$ endomorphism	every monomial automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for monomial automorphisms; i.e., it is the invariance property for monomial automorphism, which is a property stronger than the property of being an endomorphism
strong monomial automorphism $\to$ function	every strong monomial automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for strong monomial automorphisms
strong monomial automorphism $\to$ endomorphism	every strong monomial automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for strong monomial automorphisms; i.e., it is the invariance property for strong monomial automorphism, which is a property stronger than the property of being an endomorphism
strong monomial automorphism $\to$ automorphism	every strong monomial automorphism of $G$ restricts to an automorphism of $H$	the auto-invariance property for strong monomial automorphisms; i.e., it is the invariance property for strong monomial automorphism, which is a group-closed property of automorphisms

For more function restriction expressions for normality, check out Nonstandard definitions of normal subgroup#Definition via function restriction expression.

Relation implication expression

This subgroup property can be defined and viewed using a relation implication expression
View all subgroup properties having such expressions

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 $\implies$ 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.

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:

Stronger properties

The most important stronger property is characteristic subgroup. See the table below for many stronger properties and the way they're related:

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions	comparison
characteristic subgroup	invariant under all automorphisms	characteristic implies normal	normal not implies characteristic (see also list of examples)	click here	characteristic versus normal
central factor	inner automorphism of whole group is inner on subgroup	central factor implies normal	normal not implies central factor	click here	central factor versus normal
direct factor	factor in internal direct product	direct factor implies normal	normal not implies direct factor	click here	direct factor versus normal
central subgroup	contained in the center	central implies normal	normal not implies central	click here	normal subgroup versus central subgroup

[SHOW MORE]

Conjunction with other properties

Important conjunctions of normality with other subgroup properties are in the table below:[SHOW MORE]

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:[SHOW MORE]

In some cases, we are interested in studying normal subgroups with the big group constrained to satisfy some group property. For instance:[SHOW MORE]

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions	comparison
Subnormal subgroup	obtained using subordination operator on normality	normal implies subnormal	subnormal not implies normal	click here	contrasting subnormality of various depths
Permutable subgroup (also called quasinormal subgroup)	permutes with every subgroup	normal implies permutable	permutable not implies normal	click here
2-subnormal subgroup	normal subgroup of normal subgroup		normality is not transitive	click here	(above)

[SHOW MORE]

Related operators

There are three important subgroup operators related to normality:

Normal core: This takes a subgroup and outputs the largest normal subgroup inside it.
Normal closure: This takes a subgroup and outputs the smallest normal subgroup containing it.
Normalizer: This takes a subgroup and outputs the largest subgroup within which it is normal.

Other operators involve composing these in different ways, for instance:

Further information: Category: Subgroup operators related to normality

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	No	Normality is not transitive	Can have $H \le K \le G$ , $H$ normal in $K$ , $K$ is normal in $G$ , $H$ not normal in $G$ .
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	Given $H_i, i \in I$ , all normal in $G$ , so is $\bigcap_{i \in I} H_i$
Strongly join-closed subgroup property	Yes	Normality is strongly join-closed	Given $H_i, i \in I$ , all normal in $G$ , so is $\langle H_i \rangle_{i in I}$
Quotient-transitive subgroup property	Yes	Normality is quotient-transitive	$H \le K \le G$ , $H$ normal in $G$ , $K / H$ normal in $G$ , then $K$ normal in $G$
Intermediate subgroup condition	Yes	Normality satisfies intermediate subgroup condition	$H \le K \le G$ , $H$ normal in $G$ , then $H$ normal in $K$
Transfer condition	Yes	Normality satisfies transfer condition	$H, K \le G$ , $H$ normal in $G$ , then $H \cap K$ normal in $K$
Image condition	Yes	Normality satisfies image condition	$H$ normal in $G$ , $\varphi:G \to K$ surjective, then $\varphi(H)$ normal in $K$
Inverse image condition	Yes	Normality satisfies inverse image condition	$H$ normal in $G$ , $\varphi:K \to G$ homomorphism, then $\varphi^{-1}(H)$ is normal in $K$
Upper join-closed subgroup property	Yes	Normality is upper join-closed	$H \le G$ , $K_i, i \in I$ intermediate subgroups, $H$ normal in each, then $H$ is normal in their join
Commutator-closed subgroup property	Yes	Normality is commutator-closed	$H, K \le G$ both normal, then $[H, K]$ normal in $G$
Centralizer-closed subgroup property	Yes	Normality is centralizer-closed	$H$ normal in $G$ , then $C G (H)$ also normal in $G$
Direct product-closed subgroup property	Yes	Normality is direct product-closed	$H i$ normal in $G i$ implies direct product of $H i$ s normal in direct product of $G i$ s.

For more information on these metaproperties: [SHOW MORE]

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
subordination operator	normal subgroup of normal subgroup of ... of normal subgroup	subnormal subgroup
hereditarily operator	every subgroup of it is normal	hereditarily normal subgroup
upward-closure operator	every subgroup containing it is normal	upward-closed normal subgroup
maximal proper operator	proper normal subgroup contained in no other proper normal subgroup	maximal normal subgroup
minimal operator	nontrivial normal subgroup containing no other nontrivial normal subgroup	minimal normal subgroup

For more information on these property operators: [SHOW MORE]

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 commands|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);

References

Textbook references

Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, Page 6, (single definition as part of a paragraph)^{More info}
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Page 82, (formal definition, and Theorem 6 giving equivalent formulations)^{More info}. Also, Page 80 (first use).
Topics in Algebra by I. N. Herstein, Page 50, Section 2.6, (formal definition)^{More info}
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, Page 15, Proposition 1.3.15, (definition introduced through proposition)^{More info}
Algebra by Serge Lang, ISBN 038795385X, Page 14, (definition in paragraph)^{More info}
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, Page 52, Point (4.8), (formal definition, followed by equivalent definition-cum-proposition in (4.9))^{More info}

External links

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Definition links

Names in other languages:German: Normalteiler; French: Sous-groupe normal; Spanish: Subgrupo normal; Italian: Sottogruppo normale

Use Google translate to translate this page to French, German, Spanish, Italian

Facts about Normal subgroupRDF feed

Applying operator gives	Characteristic subgroup +, Transitively normal subgroup +, Subnormal subgroup +, Hereditarily normal subgroup +, Upward-closed normal subgroup +, Maximal normal subgroup +, and Minimal normal subgroup +
Defined in	Book:AlperinBell (6, ?, single definition as part of a paragraph) +, Book:DummitFoote (82, ?, formal definition, and Theorem 6 giving equivalent formulations) +, Book:Herstein (50, Section 2.6, formal definition) +, Book:RobinsonGT (15, Proposition 1.3.15, definition introduced through proposition) +, Book:Lang (14, ?, definition in paragraph) +, Book:Artin (52, Point (4.8), formal definition, followed by equivalent definition-cum-proposition in (4.9)) +, Wikipedia (?, ?, ?) +, planetmath (?, ?, ?) +, Mathworld (?, ?, ?) +, Springer Online Reference Works (?, ?, ?) +, and Citizendium (?, ?, ?) +
Defining ingredient	Kernel +, Homomorphism +, Inner automorphism +, Conjugate subgroups +, Left coset +, Right coset +, Conjugacy class +, and Commutator of two subgroups +
Dissatisfies metaproperty	Transitive subgroup property +
Left side of function restriction expression	Class-preserving automorphism +, Subgroup-conjugating automorphism +, Normal automorphism +, Weakly normal automorphism +, Monomial automorphism +, Strong monomial automorphism +, and Inner automorphism +
Page class	Term +
Quick phrase	invariant under inner automorphisms, self-conjugate subgroup +, same left and right cosets +, kernel of a homomorphism +, and subgroup that is a union of conjugacy classes +
Referenced in	Book:AlperinBell (6, ?, single definition as part of a paragraph) +, Book:DummitFoote (82, ?, formal definition, and Theorem 6 giving equivalent formulations) +, Book:Herstein (50, Section 2.6, formal definition) +, Book:RobinsonGT (15, Proposition 1.3.15, definition introduced through proposition) +, Book:Lang (14, ?, definition in paragraph) +, Book:Artin (52, Point (4.8), formal definition, followed by equivalent definition-cum-proposition in (4.9)) +, Wikipedia (?, ?, ?) +, planetmath (?, ?, ?) +, mathworld (?, ?, ?) +, Springer Online Reference Works (?, ?, ?) +, and Citizendium (?, ?, ?) +
Right side of function restriction expression	Endomorphism +, and Automorphism +
Satisfies metaproperty	Invariance property +, Endo-invariance property +, Auto-invariance property +, Abelian-tautological subgroup property +, Trim subgroup property +, Trivially true subgroup property +, Identity-true subgroup property +, Left-realized subgroup property +, Right-realized subgroup property +, Intersection-closed subgroup property +, Strongly intersection-closed subgroup property +, Join-closed subgroup property +, Strongly join-closed subgroup property +, Intermediate subgroup condition +, Transfer condition +, Inverse image condition +, Quotient-transitive subgroup property +, Image condition +, Centralizer-closed subgroup property +, Commutator-closed subgroup property +, Direct product-closed subgroup property +, First-order subgroup property +, and Function restriction-expressible subgroup property +
Stronger than	3-subnormal subgroup +, 4-subnormal subgroup +, Ascendant subgroup +, Descendant subgroup +, Serial subgroup +, Hypernormalized subgroup +, Finitarily hypernormalized subgroup +, 2-hypernormalized subgroup +, Pronormal subgroup +, Weakly pronormal subgroup +, Paranormal subgroup +, Weakly normal subgroup +, Intermediately subnormal-to-normal subgroup +, Conjugate-permutable subgroup +, Modular subgroup +, Automorph-permutable subgroup +, Elliptic subgroup +, Subnormal subgroup +, Permutable subgroup +, and 2-subnormal subgroup +
Term introduced by	Galois +
Weaker than	Fully invariant subgroup +, Strictly characteristic subgroup +, Isomorph-free subgroup +, Isomorph-containing subgroup +, Injective endomorphism-invariant subgroup +, Transitively normal subgroup +, Cocentral subgroup +, SCAB-subgroup +, Conjugacy-closed normal subgroup +, Abelian-quotient subgroup +, Normal Sylow subgroup +, Normal Hall subgroup +, Abelian normal subgroup +, Cyclic normal subgroup +, Solvable normal subgroup +, Nilpotent normal subgroup +, Simple normal subgroup +, Perfect normal subgroup +, Finite normal subgroup +, Periodic normal subgroup +, Finitely generated normal subgroup +, Normal subgroup of finite group +, Normal subgroup of finitely generated group +, Normal subgroup of periodic group +, Normal subgroup of group of prime power order +, Characteristic subgroup +, Central factor +, Direct factor +, and Central subgroup +

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
fully invariant subgroup	invariant under all endomorphisms	(via characteristic)	(via characteristic) (see also list of examples)	click here	--
strictly characteristic subgroup	invariant under surjective endomorphisms	(via characteristic)	(via characteristic)	click here
isomorph-free subgroup	no other isomorphic subgroup	(via characteristic)	(via characteristic) (see also list of examples)	click here
isomorph-containing subgroup	contains all isomorphic subgroups	(via characteristic)	(via characteristic) (see also list of examples)	click here
injective endomorphism-invariant subgroup	invariant under injective endomorphisms	(via characteristic)	(via characteristic) (see also list of examples)	click here
transitively normal subgroup	any normal subgroup is normal in whole group
cocentral subgroup	product with center is whole group	cocentral implies normal	normal not implies cocentral	click here	--
SCAB-subgroup	subgroup-conjugating automorphism restricts to subgroup-conjugating automorphism of subgroup			click here	--
conjugacy-closed normal subgroup	conjugacy-closed subgroup and normal subgroup			click here	--
abelian-quotient subgroup	contains the commutator subgroup

Conjunction	Other component of conjunction	Intermediate notions
abelian normal subgroup	abelian group	click here
* cyclic normal subgroup	cyclic group	click here
* solvable normal subgroup	solvable group
* nilpotent normal subgroup	nilpotent group	click here
simple normal subgroup	simple group
* perfect normal subgroup	perfect group	click here
finite normal subgroup	finite group	click here
* periodic normal subgroup	periodic group	click here
* finitely generated normal subgroup	finitely generated group	click here

Conjunction	Other component of conjunction	Intermediate notions
conjugacy-closed normal subgroup	conjugacy-closed subgroup	click here
transitively normal subgroup	CEP-subgroup
normal Sylow subgroup	Sylow subgroup	click here
normal Hall subgroup	Hall subgroup	click here

Conjunction	Property of the group	Intermediate notions
normal subgroup of finite group	finite group	click here
* normal subgroup of finitely generated group	finitely generated group
* normal subgroup of periodic group	periodic group
normal subgroup of group of prime power order	group of prime power order

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions	comparison
3-subnormal subgroup	normal of normal of normal			click here	contrasting subnormality of various depths
4-subnormal subgroup	2-subnormal of 2-subnormal			click here	contrasting subnormality of various depths
Ascendant subgroup				click here
Descendant subgroup				click here
Serial subgroup				click here
Hypernormalized subgroup	iterated normalizers reach whole group			click here
Finitarily hypernormalized subgroup	iterated normalizers reach whole group in finite number of steps			click here
2-hypernormalized subgroup	normalizer is normal
Pronormal subgroup	conjugate to any conjugate in their join	normal implies pronormal	pronormal not implies normal	click here	subnormal-to-normal and normal-to-characteristic
Weakly pronormal subgroup		(via pronormal)		click here	(above)
Paranormal subgroup		(via pronormal)		click here	(above)
Weakly normal subgroup		(via pronormal)		click here	(above)
Intermediately subnormal-to-normal subgroup		(via pronormal)		click here	(above)
Conjugate-permutable subgroup	permutes with its conjugate subgroups			click here
Modular subgroup				click here
Automorph-permutable subgroup				click here
Elliptic subgroup				click here

Normal subgroup

From Groupprops

Contents

Definition

Symbol-free definition

Definition with symbols

Equivalence of definitions

History

Origin of the concept

Origin of the term

Importance

Examples

Extreme examples

Examples

Non-examples

Formalisms

First-order description

Function restriction expression

Relation implication expression

Variety formalism

Relation with other properties

Stronger properties

Conjunction with other properties

Weaker properties

Related operators

Metaproperties

Contents

Tautology when whole group is abelian

Transitivity

Trimness

Intersection-closedness

Join-closedness

Intermediate subgroup condition

Transfer condition

Inverse image condition

Quotient-transitivity

Image condition

Centralizer-closedness

Commutator-closedness

Upper join-closedness

Direct product-closedness

Arguesianness

Effect of property operators

Contents

The left transiter

The right transiter

The subordination operator

The hereditarily operator

The upward-closure operator

The maximal proper operator

The minimal operator

Testing

The testing problem

GAP command

References

Textbook references

External links

Definition links

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