Algebraic structure → Group theory Group theory 

In abstract algebra, a normal subgroup (also known as an invariant subgroup or selfconjugate subgroup)^{[1]} is a subgroup that is invariant under conjugation by members of the group 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 which means that they can be used to internally classify those homomorphisms.
Évariste Galois was the first to realize the importance of the existence of normal subgroups.^{[2]}
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Normal Subgroups and Quotient Groups (aka Factor Groups)  Abstract Algebra

Abstract Algebra  Normal Subgroups

Group Theory  Normal Subgroup  Simple Group  Invariant Subgroup  Abstract Algebra

Visual Group Theory, Lecture 3.3: Normal subgroups

Why Normal Subgroups are Necessary for Quotient Groups
Transcription
Definitions
A subgroup of a group is called a normal subgroup of if it is invariant under conjugation; that is, the conjugation of an element of by an element of is always in ^{[3]} The usual notation for this relation is
Equivalent conditions
For any subgroup of the following conditions are 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 ^{[4]} i.e., for all
 The image of conjugation of by any element of is equal to ^{[4]} i.e., for all .
 For all the left and right cosets and are equal.^{[4]}
 The sets of left and right cosets of in coincide.^{[4]}
 Multiplication in preserves the equivalence relation "is in the same left coset as". That is, for every satisfying and , we have
 There exists a group on the set of left cosets of where multiplication of any two left cosets and yields the left coset . (This group is called the quotient group of modulo , denoted .)
 is a union of conjugacy classes of ^{[2]}
 is preserved by the inner automorphisms of ^{[5]}
 There is some group homomorphism whose kernel is ^{[2]}
 There exists a group homomorphism whose fibers form a group where the identity element is and multiplication of any two fibers and yields the fiber . (This group is the same group mentioned above.)
 There is some congruence relation on for which the equivalence class of the identity element is .
 For all and the commutator is in ^{[citation needed]}
 Any two elements commute modulo the normal subgroup membership relation. That is, for all if and only if ^{[citation needed]}
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.)^{[6]} 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 ^{[7]}^{[8]} More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup.^{[9]}
If is an abelian group then every subgroup of is normal, because More generally, for any group , every subgroup of the center of is normal in . (In the special case that is abelian, the center is all of , hence the fact that all subgroups of an abelian group are normal.) A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group.^{[10]}
A concrete example of a normal subgroup is the subgroup of the symmetric group consisting of the identity and both threecycles. 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 ^{[11]} This illustrates the general fact that any subgroup of index two is normal.
As an example of a normal subgroup within a matrix group, consider the general linear group of all invertible matrices with real entries under the operation of matrix multiplication and its subgroup of all matrices of determinant 1 (the special linear group). To see why the subgroup is normal in , consider any matrix in and any invertible matrix . Then using the two important identities and , one has that , and so as well. This means is closed under conjugation in , so it is a normal subgroup.^{[a]}
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.^{[12]}
The translation group is a normal subgroup of the Euclidean group in any dimension.^{[13]} 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 ^{[14]}
 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.^{[15]} However, a characteristic subgroup of a normal subgroup is normal.^{[16]} A group in which normality is transitive is called a Tgroup.^{[17]}
 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 ^{[18]}
 Normality is preserved under surjective homomorphisms;^{[19]} 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;^{[19]} 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;^{[20]} 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.^{[21]}
 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 nonidentity finite group is simple if and only if it is isomorphic to all of its nonidentity homomorphic images,^{[22]} a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is 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 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.^{[20]}
Normal subgroups, quotient groups and homomorphisms
If is a normal subgroup, we can define a multiplication on cosets as follows:
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 ^{[23]} 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 to (the first isomorphism theorem).^{[24]} 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).^{[25]} 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 ^{[26]}
See also
Operations taking subgroups to subgroups
Subgroup properties complementary (or opposite) to normality
Subgroup properties stronger than normality
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
 Cnormal subgroup
Related notions in algebra
Notes
 ^ In other language: is a homomorphism from to the multiplicative subgroup , and is the kernel. Both arguments also work over the complex numbers, or indeed over an arbitrary field.
References
 ^ Bradley 2010, p. 12.
 ^ ^{a} ^{b} ^{c} Cantrell 2000, p. 160.
 ^ Dummit & Foote 2004.
 ^ ^{a} ^{b} ^{c} ^{d} Hungerford 2003, p. 41.
 ^ Fraleigh 2003, p. 141.
 ^ Robinson 1996, p. 16.
 ^ Hungerford 2003, p. 45.
 ^ Hall 1999, p. 138.
 ^ Hall 1999, p. 32.
 ^ Hall 1999, p. 190.
 ^ Judson 2020, Section 10.1.
 ^ Bergvall et al. 2010, p. 96.
 ^ Thurston 1997, p. 218.
 ^ Hungerford 2003, p. 42.
 ^ Robinson 1996, p. 17.
 ^ Robinson 1996, p. 28.
 ^ Robinson 1996, p. 402.
 ^ Hungerford 2013, p. 290.
 ^ ^{a} ^{b} Hall 1999, p. 29.
 ^ ^{a} ^{b} Hungerford 2003, p. 46.
 ^ Robinson 1996, p. 36.
 ^ Dõmõsi & Nehaniv 2004, p. 7.
 ^ Hungerford 2003, pp. 42–43.
 ^ Hungerford 2003, p. 44.
 ^ Robinson 1996, p. 20.
 ^ Hall 1999, p. 27.
Bibliography
 Bergvall, Olof; Hynning, Elin; Hedberg, Mikael; Mickelin, Joel; Masawe, Patrick (16 May 2010). "On Rubik's Cube" (PDF). KTH.
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: Cite journal requiresjournal=
(help)  Cantrell, C.D. (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press. ISBN 9780521591805.
 Dõmõsi, Pál; Nehaniv, Chrystopher L. (2004). Algebraic Theory of Automata Networks. SIAM Monographs on Discrete Mathematics and Applications. SIAM.
 Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0471433349.
 Fraleigh, John B. (2003). A First Course in Abstract Algebra (7th ed.). AddisonWesley. ISBN 9780321156082.
 Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 9780821819678.
 Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer.
 Hungerford, Thomas (2013). Abstract Algebra: An Introduction. Brooks/Cole Cengage Learning.
 Judson, Thomas W. (2020). Abstract Algebra: Theory and Applications.
 Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). SpringerVerlag. ISBN 9781461264439. Zbl 0836.20001.
 Thurston, William (1997). Levy, Silvio (ed.). Threedimensional geometry and topology, Vol. 1. Princeton Mathematical Series. Princeton University Press. ISBN 9780691083049.
 Bradley, C. J. (2010). The mathematical theory of symmetry in solids : representation theory for point groups and space groups. Oxford New York: Clarendon Press. ISBN 9780199582587. OCLC 859155300.
Further reading
 I. N. Herstein, Topics in algebra. Second edition. Xerox College Publishing, Lexington, Mass.Toronto, Ont., 1975. xi+388 pp.
External links
 Weisstein, Eric W. "normal subgroup". MathWorld.
 Normal subgroup in Springer's Encyclopedia of Mathematics
 Robert Ash: Group Fundamentals in Abstract Algebra. The Basic Graduate Year
 Timothy Gowers, Normal subgroups and quotient groups
 John Baez, What's a Normal Subgroup?