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 *N* of the group *G* is normal in *G* if and only if *gng*^{−1} ∈ *N* for all *g* ∈ *G* and *n* ∈ *N*. 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 *G* are precisely the kernels of group homomorphisms with domain *G*, 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]}

## Definitions

A subgroup *N* of a group *G* is called a **normal subgroup** of *G* if it is invariant under conjugation; that is, the conjugation of an element of *N* by an element of *G* is always in *N*.^{[3]} The usual notation for this relation is .

### Equivalent conditions

For any subgroup *N* of *G*, the following conditions are equivalent to *N* being a normal subgroup of *G*. Therefore, any one of them may be taken as the definition:

- The image of conjugation of
*N*by any element of*G*is a subset of*N*.^{[4]} - The image of conjugation of
*N*by any element of*G*is equal to*N*.^{[4]} - For all
*g*in*G*, the left and right cosets*gN*and*Ng*are equal.^{[4]} - The sets of left and right cosets of
*N*in*G*coincide.^{[4]} - The product of an element of the left coset of
*N*with respect to*g*and an element of the left coset of*N*with respect to*h*is an element of the left coset of*N*with respect to*gh*: ∀*x*,*y*,*g*,*h*∈*G*, if*x*∈*gN*and*y*∈*hN*then*xy*∈ (*gh*)*N*. *N*is a union of conjugacy classes of*G*.^{[2]}*N*is preserved by the inner automorphisms of G.^{[5]}- There is some group homomorphism
*G*→*H*whose kernel is*N*.^{[2]} - For all and , the commutator is in N.
^{[citation needed]} - Any two elements commute regarding the normal subgroup membership relation: ∀
*g*,*h*∈*G*,*gh*∈*N*⇔*hg*∈*N*.^{[citation needed]}

## Examples

For any group G, the trivial subgroup {*e*} consisting of just the identity element of *G* is always a normal subgroup of G. Likewise, *G* itself is always a normal subgroup of *G*. (If these are the only normal subgroups, then *G* 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 *G* is an abelian group then every subgroup *N* of G is normal, because 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]}

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
*H*is a normal subgroup of*G*, and*K*is a subgroup of*G*containing*H*, then*H*is a normal subgroup of*K*.^{[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
*G*and*H*are normal subgroups of their direct product*G*×*H*. - If the group
*G*is a semidirect product then*N*is normal in*G*, though*H*need not be normal in*G*. - Normality is preserved under surjective homomorphisms,
^{[18]}i.e. if*G*→*H*is a surjective group homomorphism and*N*is normal in*G*, then the image*f*(*N*) is normal in*H*. - Normality is preserved by taking inverse images,
^{[18]}i.e. if*G*→*H*is a group homomorphism and*N*is normal in*H*, then the inverse image*f*^{ −1}(*N*) is normal in*G*. - Normality is preserved on taking direct products,
^{[19]}i.e. if and , then . - Every subgroup of index 2 is normal. More generally, a subgroup,
*H*, of finite index,*n*, in*G*contains a subgroup,*K*, normal in*G*and of index dividing*n*! called the normal core. In particular, if*p*is the smallest prime dividing the order of*G*, then every subgroup of index*p*is normal.^{[20]} - The fact that normal subgroups of
*G*are precisely the kernels of group homomorphisms defined on*G*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,^{[21]}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, *N* and *M*, of *G*, their intersection and their product are also normal subgroups of *G*.

The normal subgroups of *G* form a lattice under subset inclusion with least element, {*e*} , and greatest element, *G*. The meet of two normal subgroups, *N* and *M*, in this lattice is their intersection and the join is their product.

The lattice is complete and modular.^{[19]}

## Normal subgroups, quotient groups and homomorphisms

If *N* is a normal subgroup, we can define a multiplication on cosets as follows:

*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 *G*/*N*. There is a natural homomorphism, *f*: *G* → *G/N*, given by *f*(*a*) = *aN*. This homomorphism maps into the identity element of *G/N*, which is the coset *eN* = *N*,^{[22]} that is, .

In general, a group homomorphism, *f*: *G* → *H* sends subgroups of *G* to subgroups of *H*. Also, the preimage of any subgroup of *H* is a subgroup of *G*. We call the preimage of the trivial group {*e*} in *H* the **kernel** of the homomorphism and denote it by ker(*f*). As it turns out, the kernel is always normal and the image of *G*, *f*(*G*), is always isomorphic to *G*/ker(*f*) (the first isomorphism theorem).^{[23]} In fact, this correspondence is a bijection between the set of all quotient groups of G, *G*/*N*, and the set of all homomorphic images of *G* (up to isomorphism).^{[24]} It is also easy to see that the kernel of the quotient map, *f*: *G* → *G/N*, is *N* itself, so the normal subgroups are precisely the kernels of homomorphisms with domain *G*.^{[25]}

## 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
- C-normal subgroup

### Related notions in algebra

## Notes

**^**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.- ^
^{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.

## References

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*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. - 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.**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.