In
abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a
subgroup
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
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 change ...
by members of the
group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
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 (mathematics), 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 "factore ...
s of the given group. Furthermore, the normal subgroups of
are precisely the
kernels of
group homomorphismsImage:Group homomorphism ver.2.svg, 250px, Image of a group homomorphism (h) from G (left) to H (right). The smaller oval inside H is the image of h. N is the Kernel_(algebra)#Group_homomorphisms, kernel of h and aN is a coset of N.
In mathematics, ...

with domain
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
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ...
was the first to realize the importance of the existence of normal subgroups.
Definitions
A
subgroup
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
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 change ...
; 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
*''Equivalent ...
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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 of
conjugacy class
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
es of
*
is preserved by the
inner automorphism
In abstract algebra an inner automorphism is an automorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...
s of
* There is some
group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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
* For all
and
the
commutator
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is in
* Any two elements commute regarding the normal subgroup membership relation: 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 group, simple.) Other named normal subgroups of an arbitrary group include the Center (group theory), center of the group (the set of elements that commute with all other elements) and the commutator subgroup
More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup.
If
is an abelian group 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.
A concrete example of a normal subgroup is the subgroup
of the symmetric group
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 subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal.
The translation group 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 Rotation, 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. The smallest group exhibiting this phenomenon is the dihedral group of order 8. However, a characteristic subgroup of a normal subgroup is normal. A group in which normality is transitive is called a T-group (mathematics), T-group.
* The two groups
and
are normal subgroups of their Direct product of groups, direct product
* If the group
is a semidirect product
then
is normal in
though
need not be normal in
* 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 image, 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 product of groups, direct products; that is, if
and
then
* Every subgroup of Index (group theory), index 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 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 group, simple if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is Perfect group, perfect if and only if it has no normal subgroups of prime Index of a subgroup, index, and a group is Imperfect group, imperfect if and only if the derived subgroup 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 (order), lattice under subset inclusion with least element,
and greatest element,
The Meet (lattice theory), meet of two normal subgroups,
and
in this lattice is their intersection and the Join (lattice theory), join is their product.
The lattice is Complete lattice, complete and Modular lattice, modular.
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 (mathematics), 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 "factore ...
and denoted with
There is a natural Group homomorphism, homomorphism,
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 to
(the first isomorphism theorem). 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 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 of a function, domain
See also
Operations taking subgroups to subgroups
*Normalizer
*Conjugate closure
*Normal core
Subgroup properties complementary (or opposite) to normality
*Malnormal subgroup
*Contranormal subgroup
*Abnormal subgroup
*Self-normalizing subgroup
Subgroup properties stronger than normality
*Characteristic subgroup
*Fully characteristic subgroup
Subgroup properties weaker than normality
*Subnormal subgroup
*Ascendant subgroup
*Descendant subgroup
*Quasinormal subgroup
*Seminormal subgroup
*Conjugate permutable subgroup
*Modular subgroup
*Pronormal subgroup
*Paranormal subgroup
*Polynormal subgroup
*C-normal subgroup
Related notions in algebra
*Ideal (ring theory)
Notes
References
*
*
*
*
*
*
*
*
*
*
*
Further reading
* I. N. Herstein, ''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