# Normal subgroup

A **normal subgroup** of a group G is a subgroup H for which the left and right cosets coincide: aH = Ha for every element a of G, equivalently gHg⁻¹ = H for every g in G. Such a subgroup is also called an invariant or self-conjugate subgroup, and the relation is written H ⊴ G (or H ⋬ G when H is a proper subgroup)<sup>[1](https://encyclopediaofmath.org/wiki/Normal_subgroup)</sup>. Normality is the exact condition under which the cosets of H can themselves be made into a group, the quotient group G/H, and it is precisely the class of subgroups that arise as kernels of homomorphisms<sup>[2](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_%28Denton%29/05%3A_Groups_IV/5.01%3A_Quotient_Groups)</sup>.

| Key fact | Detail |
|---|---|
| Definition | H ⊴ G if aH = Ha for all a ∈ G, equivalently gHg⁻¹ = H for all g ∈ G<sup>[1](https://encyclopediaofmath.org/wiki/Normal_subgroup)</sup> |
| Conjugation form | H is normal if and only if it contains all G-conjugates of each of its elements, i.e. it is a union of conjugacy classes of G<sup>[1](https://encyclopediaofmath.org/wiki/Normal_subgroup)</sup><sup> • </sup><sup>[3](https://sites.ualberta.ca/~vbouchar/MAPH464/section-normal.html)</sup>  |
| Kernels | A subgroup of G is normal if and only if it is the kernel of a homomorphism<sup>[2](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_%28Denton%29/05%3A_Groups_IV/5.01%3A_Quotient_Groups)</sup> |
| Quotient | If N ⊴ G, the cosets form a group G/N under (aN)(bN) = abN, of order [G : N]<sup>[4](https://ntouikan.ext.unb.ca/MATH4043/groups-section-homomorphisms-normal-quotients.html)</sup> |
| First isomorphism theorem | For a homomorphism θ : G → G′, the image θ(G) is isomorphic to G/ker θ<sup>[5](https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/14%3A_Group_Theory_and_Applications/14.04%3A_Normal_Subgroups_and_Group_Homomorphisms)</sup> |
| Always normal | Every subgroup of an abelian group, the center Z(G), the alternating group Aₙ inside Sₙ, and any subgroup of index 2<sup>[3](https://sites.ualberta.ca/~vbouchar/MAPH464/section-normal.html)</sup><sup> • </sup><sup>[6](https://crypto.stanford.edu/pbc/notes/group/normal.html)</sup>  |
| History | Évariste Galois, by 1832, identified these special subgroups as fundamental, calling a coset decomposition proper when the left and right decompositions coincide<sup>[7](https://mathshistory.st-andrews.ac.uk/HistTopics/Development_group_theory/)</sup> |

## Equivalent definitions

The several characterizations of normality are interchangeable. A subgroup H is normal in G if and only if any one of the following holds<sup>[1](https://encyclopediaofmath.org/wiki/Normal_subgroup)</sup><sup> • </sup><sup>[3](https://sites.ualberta.ca/~vbouchar/MAPH464/section-normal.html)</sup><sup> • </sup><sup>[5](https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/14%3A_Group_Theory_and_Applications/14.04%3A_Normal_Subgroups_and_Group_Homomorphisms)</sup>:

1. aH = Ha for every a ∈ G (left and right cosets agree as sets).
2. gHg⁻¹ = H for every g ∈ G; it is enough to check the inclusion gHg⁻¹ ⊆ H for all g, since the reverse inclusion then follows.
3. H contains all G-conjugates of each of its elements, so H is a union of conjugacy classes of G.
4. The operation on left cosets induced by the operation of G, (aH)(bH) = abH, is well defined, meaning independent of the chosen coset representatives.

The conjugacy-class view gives a practical test: a subgroup is normal exactly when it "does not break conjugacy classes"<sup>[3](https://sites.ualberta.ca/~vbouchar/MAPH464/section-normal.html)</sup>. In S₄, for example, the union of the identity class with the class {(12)(34), (13)(24), (14)(23)} forms the subgroup V, which is therefore normal<sup>[6](https://crypto.stanford.edu/pbc/notes/group/normal.html)</sup>.

Normality is a relation between a subgroup and the ambient group, not a property of the subgroup alone: a subgroup H may be normal in one group and not normal in a larger group containing it<sup>[8](https://netmaths.in/book/complex-analysis-algebra-topology/groups/normal-subgroups-quotients-isomorphism-theorems)</sup>.

## Kernels of homomorphisms

For any homomorphism φ : G → G′, the kernel, the set of elements mapped to the identity of G′, is a normal subgroup of G. The proof is short: if φ(k) = e′ and g ∈ G, then φ(gkg⁻¹) = φ(g)φ(k)φ(g)⁻¹ = φ(g)e′φ(g)⁻¹ = e′, so gkg⁻¹ lies in the kernel<sup>[5](https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/14%3A_Group_Theory_and_Applications/14.04%3A_Normal_Subgroups_and_Group_Homomorphisms)</sup>.

The converse also holds, and this is the structural reason normality matters. Given N ⊴ G, the canonical map φ : G → G/N sending g to gN is a homomorphism with kernel exactly N, since φ(g₁g₂) = g₁g₂N = (g₁N)(g₂N)<sup>[4](https://ntouikan.ext.unb.ca/MATH4043/groups-section-homomorphisms-normal-quotients.html)</sup>. Hence normal subgroups and kernels of homomorphisms are the same class of objects<sup>[2](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_%28Denton%29/05%3A_Groups_IV/5.01%3A_Quotient_Groups)</sup>. Standard examples include SLₙ(F) as the kernel of the determinant map det : GLₙ(F) → F×, and Aₙ as the kernel of the sign homomorphism sgn : Sₙ → ℤ₂<sup>[9](https://www.math.wustl.edu/~matkerr/5031/A1IIi.pdf)</sup>.

## Quotient groups

For a normal subgroup N of G, the set G/N of cosets carries a group structure under (aN)(bN) = abN. Normality is what makes this multiplication well defined: if aN = bN and cN = dN, writing a = bn₁ and using normality to move n₁ past d gives abN = cdN, so the product does not depend on the representatives chosen<sup>[4](https://ntouikan.ext.unb.ca/MATH4043/groups-section-homomorphisms-normal-quotients.html)</sup>. The identity of G/N is the coset N itself, the inverse of gN is g⁻¹N, associativity is inherited from G, and when G is finite the order of G/N equals the index [G : N]<sup>[4](https://ntouikan.ext.unb.ca/MATH4043/groups-section-homomorphisms-normal-quotients.html)</sup>. For non-normal subgroups the cosets cannot be given a compatible group structure in this way, which is why quotients exist precisely for normal subgroups<sup>[8](https://netmaths.in/book/complex-analysis-algebra-topology/groups/normal-subgroups-quotients-isomorphism-theorems)</sup>.

Familiar quotients illustrate the construction. Since ℤ is abelian, nℤ is normal and ℤ/nℤ is the group of residue classes modulo n<sup>[9](https://www.math.wustl.edu/~matkerr/5031/A1IIi.pdf)</sup>. The quotient Sₙ/Aₙ has two elements and is isomorphic to ℤ₂<sup>[9](https://www.math.wustl.edu/~matkerr/5031/A1IIi.pdf)</sup>. The map φ(r) = e^(2πir) from ℝ to the circle group S¹ has kernel ℤ, so ℝ/ℤ ≅ S¹<sup>[9](https://www.math.wustl.edu/~matkerr/5031/A1IIi.pdf)</sup>.

The natural map a ↦ aN also has a universal property: any homomorphism α : G → G′ with α(N) = {e} factors uniquely through G/N<sup>[10](https://math.libretexts.org/Workbench/Group_Theory_4e_%28Milne%29/01%3A_Basic_Definitions_and_Results/1.08%3A_Kernels_and_quotients)</sup>.

## The first isomorphism theorem

The three ideas of homomorphism, kernel, and quotient are tied together by the fundamental theorem of [group homomorphisms](https://www.edgepedia.org/group-homomorphism): if θ : G → G′ is a homomorphism with kernel K, then K is normal in G and the map gK ↦ θ(g) is an isomorphism from G/K onto the image θ(G)<sup>[5](https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/14%3A_Group_Theory_and_Applications/14.04%3A_Normal_Subgroups_and_Group_Homomorphisms)</sup><sup> • </sup><sup>[9](https://www.math.wustl.edu/~matkerr/5031/A1IIi.pdf)</sup>. In particular, for a surjective homomorphism, G/ker θ ≅ G′, and when G is finite, |G|/|K| = |θ(G)|<sup>[9](https://www.math.wustl.edu/~matkerr/5031/A1IIi.pdf)</sup>. The theorem is among the most-used results in algebra<sup>[11](https://math.hodapp.com/lectures/foundations/groups/homomorphisms-and-quotients.html)</sup>.

A further correspondence follows: for K ⊴ G, the map H ↦ H/K is a bijection between the subgroups of G containing K and the subgroups of G/K, and H is normal in G exactly when H/K is normal in G/K, in which case G/H ≅ (G/K)/(H/K)<sup>[9](https://www.math.wustl.edu/~matkerr/5031/A1IIi.pdf)</sup>.

## Closure properties and examples

Normal subgroups behave well under the standard constructions. The intersection of any set of normal subgroups of G is normal, and the subgroup generated by any system of normal subgroups is normal in G<sup>[1](https://encyclopediaofmath.org/wiki/Normal_subgroup)</sup>.

Classes of subgroups that are always normal include<sup>[3](https://sites.ualberta.ca/~vbouchar/MAPH464/section-normal.html)</sup><sup> • </sup><sup>[6](https://crypto.stanford.edu/pbc/notes/group/normal.html)</sup>:

- Every subgroup of an abelian group, since ghg⁻¹ = h for all elements.
- The center Z(G), whose elements commute with everything in G.
- The alternating group Aₙ inside the symmetric group Sₙ, for every n.
- Any subgroup of index 2: the two cosets H and sH = Hs force H to commute with every element of G.
- In the [dihedral group](https://www.edgepedia.org/dihedral-group) D₂ₙ, the cyclic subgroup ⟨a⟩ of rotations.

A group with no proper normal subgroups is called **simple**. Cyclic groups of prime order are simple, and simple groups of composite order are comparatively rare<sup>[12](https://crypto.stanford.edu/pbc/notes/group/jordanholder.html)</sup>. A proper normal subgroup A is a maximum normal subgroup of G exactly when G/A is simple<sup>[12](https://crypto.stanford.edu/pbc/notes/group/jordanholder.html)</sup>.

Normality sits below stronger invariance notions: a subgroup invariant under all automorphisms of G is called characteristic, and one invariant under all endomorphisms is fully characteristic<sup>[1](https://encyclopediaofmath.org/wiki/Normal_subgroup)</sup>.

## Role in solvability and composition theory

Normal subgroups are the links in the chains used to measure how a group is built. A subnormal series of G is a finite sequence of subgroups H₀, H₁, …, Hₙ with H₀ = {e}, Hₙ = G, and each Hᵢ normal in Hᵢ₊₁<sup>[13](https://people.utm.my/nizasarmin/wp-content/uploads/sites/980/2021/09/Full-Lecture-Notes-GT1-copyrighted.pdf)</sup>. A composition series is such a chain in which every successive quotient is a simple group<sup>[12](https://crypto.stanford.edu/pbc/notes/group/jordanholder.html)</sup>.

Solvability is defined through such chains: a group is solvable if it has a finite normal series with abelian successive quotients, and a finite group G with a normal subgroup H is solvable if and only if both H and G/H are solvable<sup>[14](https://maths.dur.ac.uk/users/daniel.evans/GaloisTheory/Notes/GaloisTheoryNotes.pdf)</sup><sup> • </sup><sup>[17](https://encyclopediaofmath.org/wiki/Solvable_group)</sup>. This is the group-theoretic content behind Galois' criterion that a polynomial is solvable by radicals exactly when its [Galois group](https://www.edgepedia.org/galois-group) is solvable<sup>[15](https://math.hodapp.com/lectures/foundations/fields/galois.html)</sup>.

## History

The concept originates with [Évariste Galois](https://www.edgepedia.org/evariste-galois). By 1832 he had discovered that special subgroups, now called normal subgroups, are fundamental, and he called the decomposition of a group into cosets of a subgroup a proper decomposition when the right and left coset decompositions coincide<sup>[7](https://mathshistory.st-andrews.ac.uk/HistTopics/Development_group_theory/)</sup>. A [history of algebra](https://www.edgepedia.org/history-of-algebra) credits Galois in 1832 with defining the concept of a normal subgroup and of a solvable group<sup>[16](https://math.hawaii.edu/~lee/algebra/history.html)</sup>. Galois' work remained unpublished and unknown until Joseph Liouville published his papers in 1846<sup>[7](https://mathshistory.st-andrews.ac.uk/HistTopics/Development_group_theory/)</sup>.

## References

1. [Normal subgroup, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Normal_subgroup)
2. [5.1: Quotient Groups, Mathematics LibreTexts (Denton)](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_%28Denton%29/05%3A_Groups_IV/5.01%3A_Quotient_Groups)
3. [Normal subgroups, MAPH464 notes, University of Alberta](https://sites.ualberta.ca/~vbouchar/MAPH464/section-normal.html)
4. [AATA: Homomorphisms, Normal Subgroups and Quotients, University of New Brunswick](https://ntouikan.ext.unb.ca/MATH4043/groups-section-homomorphisms-normal-quotients.html)
5. [14.4: Normal Subgroups and Group Homomorphisms, Mathematics LibreTexts](https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/14%3A_Group_Theory_and_Applications/14.04%3A_Normal_Subgroups_and_Group_Homomorphisms)
6. [Group Theory: Normal Subgroups, Stanford notes](https://crypto.stanford.edu/pbc/notes/group/normal.html)
7. [Development of group theory, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Development_group_theory/)
8. [Normal subgroups, quotients, isomorphism theorems, NETMaths](https://netmaths.in/book/complex-analysis-algebra-topology/groups/normal-subgroups-quotients-isomorphism-theorems)
9. [II.I. Normal Subgroups and Quotient Groups, Washington University lecture notes](https://www.math.wustl.edu/~matkerr/5031/A1IIi.pdf)
10. [1.8: Kernels and quotients, Milne, Group Theory, Mathematics LibreTexts](https://math.libretexts.org/Workbench/Group_Theory_4e_%28Milne%29/01%3A_Basic_Definitions_and_Results/1.08%3A_Kernels_and_quotients)
11. [Homomorphisms, normal subgroups, and quotient groups, Math foundations](https://math.hodapp.com/lectures/foundations/groups/homomorphisms-and-quotients.html)
12. [Group Theory: Jordan–Hölder Decomposition, Stanford notes](https://crypto.stanford.edu/pbc/notes/group/jordanholder.html)
13. [Full Lecture Notes on Group Theory, Universiti Teknologi Malaysia](https://people.utm.my/nizasarmin/wp-content/uploads/sites/980/2021/09/Full-Lecture-Notes-GT1-copyrighted.pdf)
14. [Galois Theory III (MATH3041), Durham University](https://maths.dur.ac.uk/users/daniel.evans/GaloisTheory/Notes/GaloisTheoryNotes.pdf)
15. [Galois theory: a panorama, Math foundations](https://math.hodapp.com/lectures/foundations/fields/galois.html)
16. [The Origins of Modern Algebra, University of Hawaii](https://math.hawaii.edu/~lee/algebra/history.html)
17. [encyclopediaofmath.org](https://encyclopediaofmath.org/wiki/Solvable_group)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Group theory › Group structures and subgroups*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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