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 subgro ...
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 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 groups of the given group. Furthermore, the normal subgroups of
are precisely the
kernels of
group 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
* ...
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 subgro ...
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
*'' Equiv ...
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
cosets 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 classes 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 whose
kernel is
* There is some
congruence relation on
for which the
equivalence class 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 ...
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 More generally, since conjugation is an isomorphism, any
characteristic subgroup 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 commu ...
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
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 grou ...
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
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 mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations ...
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
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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 ho ...
. 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, ...
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.
* The two groups
and
are normal subgroups of their
direct product
* If the group
is a
semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in ...
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 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
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, 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 ...
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 under
subset inclusion with
least element,
and
greatest element,
The
meet of two normal subgroups,
and
in this lattice is their intersection and the
join is their product.
The lattice is
complete 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 sy ...
.
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 and denoted with
There is a natural
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 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 iso ...
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 Two 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'' with respect to ''R'' ...
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
* ...
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
*
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