In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate 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]



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]



For any group  , the trivial subgroup   consisting of only 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 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  .[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.


  • 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 T-group.[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 non-identity finite group is simple if and only if it is isomorphic to all of its non-identity 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:

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 normal 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  ,[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



  1. ^ 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.


  1. ^ Bradley 2010, p. 12.
  2. ^ a b c Cantrell 2000, p. 160.
  3. ^ Dummit & Foote 2004.
  4. ^ a b c d Hungerford 2003, p. 41.
  5. ^ Fraleigh 2003, p. 141.
  6. ^ Robinson 1996, p. 16.
  7. ^ Hungerford 2003, p. 45.
  8. ^ Hall 1999, p. 138.
  9. ^ Hall 1999, p. 32.
  10. ^ Hall 1999, p. 190.
  11. ^ Judson 2020, Section 10.1.
  12. ^ Bergvall et al. 2010, p. 96.
  13. ^ Thurston 1997, p. 218.
  14. ^ Hungerford 2003, p. 42.
  15. ^ Robinson 1996, p. 17.
  16. ^ Robinson 1996, p. 28.
  17. ^ Robinson 1996, p. 402.
  18. ^ Hungerford 2013, p. 290.
  19. ^ a b Hall 1999, p. 29.
  20. ^ a b Hungerford 2003, p. 46.
  21. ^ Robinson 1996, p. 36.
  22. ^ Dõmõsi & Nehaniv 2004, p. 7.
  23. ^ Hungerford 2003, pp. 42–43.
  24. ^ Hungerford 2003, p. 44.
  25. ^ Robinson 1996, p. 20.
  26. ^ Hall 1999, p. 27.


  • Bergvall, Olof; Hynning, Elin; Hedberg, Mikael; Mickelin, Joel; Masawe, Patrick (16 May 2010). "On Rubik's Cube" (PDF). KTH.
  • Cantrell, C.D. (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press. ISBN 978-0-521-59180-5.
  • 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 0-471-43334-9.
  • Fraleigh, John B. (2003). A First Course in Abstract Algebra (7th ed.). Addison-Wesley. ISBN 978-0-321-15608-2.
  • Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8.
  • 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.). Springer-Verlag. ISBN 978-1-4612-6443-9. Zbl 0836.20001.
  • Thurston, William (1997). Levy, Silvio (ed.). Three-dimensional geometry and topology, Vol. 1. Princeton Mathematical Series. Princeton University Press. ISBN 978-0-691-08304-9.
  • 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 978-0-19-958258-7. OCLC 859155300.

Further reading

  • I. N. Herstein, Topics in algebra. Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp.