In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
that is invariant under
conjugation
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the chang ...
by members of the
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of which it is a part. In other words, a subgroup
of the group
is normal in
if and only if
for all
and
The usual notation for this relation is
Normal subgroups are important because they (and only they) can be used to construct
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
s of the given group. Furthermore, the normal subgroups of
are precisely the
kernels
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
of
group homomorphisms
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
...
with
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
which means that they can be used to internally classify those homomorphisms.
Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
was the first to realize the importance of the existence of normal subgroups.
Definitions
A
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of a group
is called a normal subgroup of
if it is invariant under
conjugation
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the chang ...
; that is, the conjugation of an element of
by an element of
is always in
The usual notation for this relation is
Equivalent conditions
For any subgroup
of
the following conditions are
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
to
being a normal subgroup of
Therefore, any one of them may be taken as the definition:
* The image of conjugation of
by any element of
is a subset of
* The image of conjugation of
by any element of
is equal to
* For all
the left and right cosets
and
are equal.
* The sets of left and right
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s of
in
coincide.
* The product of an element of the left coset of
with respect to
and an element of the left coset of
with respect to
is an element of the left coset of
with respect to
: for all
if
and
then
*
is a
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of
conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...
es of
*
is preserved by the
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group it ...
s of
* There is some
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
wh ...
whose
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
is
* There is some
congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...
on
for which the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
is
.
* For all
and
the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
is in
* Any two elements commute regarding the normal subgroup membership relation. That is, for all
if and only if
Examples
For any group
the trivial subgroup
consisting of just the identity element of
is always a normal subgroup of
Likewise,
itself is always a normal subgroup of
(If these are the only normal subgroups, then
is said to be
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
.) Other named normal subgroups of an arbitrary group include the
center of the group (the set of elements that commute with all other elements) and the
commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
More generally, since conjugation is an isomorphism, any
characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphi ...
is a normal subgroup.
If
is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
then every subgroup
of
is normal, because
A group that is not abelian but for which every subgroup is normal is called a
Hamiltonian group In group theory, a Dedekind group is a group ''G'' such that every subgroup of ''G'' is normal.
All abelian groups are Dedekind groups.
A non-abelian Dedekind group is called a Hamiltonian group.
The most familiar (and smallest) example of a Hamilt ...
.
A concrete example of a normal subgroup is the subgroup
of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
consisting of the identity and both three-cycles. In particular, one can check that every coset of
is either equal to
itself or is equal to
On the other hand, the subgroup
is not normal in
since
This illustrates the general fact that any subgroup
of index two is normal.
In the
Rubik's Cube group
The Rubik's Cube group is a Group (mathematics), group (G, \cdot ) that represents the Mathematical structure, structure of the Rubik's Cube mechanical puzzle. Each element of the Set (mathematics), set G corresponds to a cube move, which is the ...
, the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal.
The
translation group
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every ...
is a normal subgroup of the
Euclidean group in any dimension. This means: applying a rigid transformation, followed by a translation and then the inverse rigid transformation, has the same effect as a single translation. By contrast, the subgroup of all
rotations about the origin is ''not'' a normal subgroup of the Euclidean group, as long as the dimension is at least 2: first translating, then rotating about the origin, and then translating back will typically not fix the origin and will therefore not have the same effect as a single rotation about the origin.
Properties
* If
is a normal subgroup of
and
is a subgroup of
containing
then
is a normal subgroup of
* A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a
transitive relation
In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive.
Definition
A homog ...
. The smallest group exhibiting this phenomenon is the
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
of order 8. However, a
characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphi ...
of a normal subgroup is normal. A group in which normality is transitive is called a
T-group.
* The two groups
and
are normal subgroups of their
direct product
* If the group
is a
semidirect product then
is normal in
though
need not be normal in
* If
and
are normal subgroups of an additive group
such that
and
, then
* Normality is preserved under surjective homomorphisms; that is, if
is a surjective group homomorphism and
is normal in
then the image
is normal in
* Normality is preserved by taking
inverse images; that is, if
is a group homomorphism and
is normal in
then the inverse image
is normal in
* Normality is preserved on taking
direct products; that is, if
and
then
* Every subgroup of
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
2 is normal. More generally, a subgroup,
of finite index,
in
contains a subgroup,
normal in
and of index dividing
called the
normal core In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the ''p''-core of a group.
The normal core Definition
For a group ''G'', the n ...
. In particular, if
is the smallest prime dividing the order of
then every subgroup of index
is normal.
* The fact that normal subgroups of
are precisely the kernels of group homomorphisms defined on
accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a non-identity finite group is
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is
perfect if and only if it has no normal subgroups of prime
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
, and a group is
imperfect
The imperfect ( abbreviated ) is a verb form that combines past tense (reference to a past time) and imperfective aspect (reference to a continuing or repeated event or state). It can have meanings similar to the English "was walking" or "used to ...
if and only if the
derived subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
is not supplemented by any proper normal subgroup.
Lattice of normal subgroups
Given two normal subgroups,
and
of
their intersection
and their product
are also normal subgroups of
The normal subgroups of
form a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
under
subset inclusion
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset ...
with
least element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an eleme ...
,
and
greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an eleme ...
,
The
meet
Meet may refer to:
People with the name
* Janek Meet (born 1974), Estonian footballer
* Meet Mukhi (born 2005), Indian child actor
Arts, entertainment, and media
* ''Meet'' (TV series), an early Australian television series which aired on ABC du ...
of two normal subgroups,
and
in this lattice is their intersection and the
join Join may refer to:
* Join (law), to include additional counts or additional defendants on an indictment
*In mathematics:
** Join (mathematics), a least upper bound of sets orders in lattice theory
** Join (topology), an operation combining two top ...
is their product.
The lattice is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and
modular
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
.
Normal subgroups, quotient groups and homomorphisms
If
is a normal subgroup, we can define a multiplication on cosets as follows:
This relation defines a mapping
To show that this mapping is well-defined, one needs to prove that the choice of representative elements
does not affect the result. To this end, consider some other representative elements
Then there are
such that
It follows that
where we also used the fact that
is a subgroup, and therefore there is
such that
This proves that this product is a well-defined mapping between cosets.
With this operation, the set of cosets is itself a group, called the
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
and denoted with
There is a natural
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
,
given by
This homomorphism maps
into the identity element of
which is the coset
that is,
In general, a group homomorphism,
sends subgroups of
to subgroups of
Also, the preimage of any subgroup of
is a subgroup of
We call the preimage of the trivial group
in
the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
of the homomorphism and denote it by
As it turns out, the kernel is always normal and the image of
is always
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to
(the
first isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
). In fact, this correspondence is a bijection between the set of all quotient groups of
and the set of all homomorphic images of
(
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
isomorphism). It is also easy to see that the kernel of the quotient map,
is
itself, so the normal subgroups are precisely the kernels of homomorphisms with
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
Normal subgroups and Sylow Theorem
The Second
Sylow Theorem states: If
and
are
Sylow p-subgroups of a group
, then there exists
such that
There is a direct corollary of the theorem above:
Let
be a finite group and
a Sylow p-subgroup for some prime
. Then
is normal in
if and only if
is the only Sylow p-subgroup in
.
See also
Operations taking subgroups to subgroups
*
Normalizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
*
Conjugate closure
In group theory, the normal closure of a subset S of a group G is the smallest normal subgroup of G containing S.
Properties and description
Formally, if G is a group and S is a subset of G, the normal closure \operatorname_G(S) of S is the ...
*
Normal core In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the ''p''-core of a group.
The normal core Definition
For a group ''G'', the n ...
Subgroup properties complementary (or opposite) to normality
*
Malnormal subgroup In mathematics, in the field of group theory, a subgroup H of a group G is termed malnormal if for any x in G but not in H, H and xHx^ intersect in the identity element
In mathematics, an identity element, or neutral element, of a binary operation ...
*
Contranormal subgroup
In mathematics, in the field of group theory, a contranormal subgroup is a subgroup whose
normal closure in the group is the whole group. Clearly, a contranormal subgroup can be normal only if it is the whole group.
Some facts:
* Every subgroup ...
*
Abnormal subgroup In mathematics, specifically group theory, an abnormal subgroup is a subgroup ''H'' of a group ''G'' such that for all ''x'' in ''G'', ''x'' lies in the subgroup generated by ''H'' and ''H'x'', where ''H'x'' denotes the conjugate subgroup
...
*
Self-normalizing subgroup
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group (mathematics), group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutativity, com ...
Subgroup properties stronger than normality
*
Characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphi ...
*
Fully characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphis ...
Subgroup properties weaker than normality
*
Subnormal subgroup
*
Ascendant subgroup
In mathematics, in the field of group theory, a subgroup of a group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural id ...
*
Descendant subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predec ...
*
Quasinormal subgroup __NOTOC__
In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups. The term ''quasinormal su ...
*
Seminormal subgroup In mathematics, in the field of group theory, a subgroup A of a group G is termed seminormal if there is a subgroup B such that AB = G, and for any proper subgroup C of B, AC is a proper subgroup of G.
This definition of seminormal subgroups is du ...
*
Conjugate permutable subgroup In mathematics, in the field of group theory, a conjugate-permutable subgroup is a subgroup that commutes with all its conjugate subgroups. The term was introduced by Tuval Foguel in 1997. and arose in the context of the proof that for finite gro ...
*
Modular subgroup
*
Pronormal subgroup In mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroups and abnormal subgroups such as Sylow subgroups, .
...
*
Paranormal subgroup
In mathematics, in the field of group theory, a paranormal subgroup is a subgroup such that the subgroup generated by it and any conjugate of it, is also generated by it and a conjugate of it ''within'' that subgroup.
In symbols, H is paranorma ...
*
Polynormal subgroup
In mathematics, in the field of group theory, a subgroup of a group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural i ...
*
C-normal subgroup
In mathematics, in the field of group theory, a subgroup H of a group G is called c-normal if there is a normal subgroup T of G such that HT = G and the intersection of H and T lies inside the normal core In group theory, a branch of mathematics ...
Related notions in algebra
*
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers pr ...
Notes
References
*
*
*
*
*
*
*
*
*
*
*
*
Further reading
*
I. N. Herstein
Israel Nathan Herstein (March 28, 1923 – February 9, 1988) was a mathematician, appointed as professor at the University of Chicago in 1951. He worked on a variety of areas of algebra, including ring theory, with over 100 research papers and ov ...
, ''Topics in algebra.'' Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp.
External links
* {{MathWorld, urlname=NormalSubgroup, title= normal subgroup
Normal subgroup in Springer's Encyclopedia of MathematicsRobert Ash: Group Fundamentals in ''Abstract Algebra. The Basic Graduate Year''Timothy Gowers, Normal subgroups and quotient groups
Subgroup properties