Normal subgroup

Lore & Background

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. The usual notation for this relation is N ◃ G. For any subgroup N of G, several conditions are equivalent to N being a normal subgroup, including that the image of conjugation of N by any element of G is a subset of N, or equal to N, or that left and right cosets gN and Ng are equal for all g in G.

Reader's Guide

Normal subgroups are fundamental in group theory because they allow the construction of quotient groups, denoted G/N, which are groups on the set of left cosets of N. This construction is possible only when N is normal. Additionally, normal subgroups are exactly the kernels of group homomorphisms from G, providing a way to classify homomorphisms internally. Évariste Galois was the first to realize the importance of the existence of normal subgroups. The concept is also known as invariant subgroup or self-conjugate subgroup, and it is defined by the condition that for all g in G and n in N, the element gng⁻¹ is in N.

Did You Know?

• A normal subgroup is also known as an invariant subgroup or self-conjugate subgroup.
• Normal subgroups are precisely the kernels of group homomorphisms with domain G.
• Évariste Galois was the first to realize the importance of the existence of normal subgroups.
• The usual notation for a normal subgroup N of G is N ◃ G.

Definition and the Conjugation Criterion

A normal subgroup—sometimes called an invariant subgroup or a self-conjugate subgroup—occupies a special position within a larger group. The defining requirement is invariance under conjugation: if N is a subgroup of a group G, then N is normal precisely when, for every element g drawn from G and every element n drawn from N, the conjugate gng⁻¹ still belongs to N. In other words, no matter which element of the ambient group you use to sandwich an element of N, the result never escapes N. This single condition is so central that it serves as the standard definition, and mathematicians record the relationship with the compact notation N ◃ G, where the triangular symbol points toward the larger group. The concept generalizes the idea that certain substructures remain stable even when the surrounding algebraic environment acts on them, making normality a measure of how well a subgroup fits inside its parent group.

Multiple Faces of Normality

One of the most elegant features of normal subgroups is that the single conjugation criterion can be restated in several equivalent ways, any of which may serve as the working definition. For instance, the condition gNg⁻¹ ⊆ N for every g in G can be strengthened to equality, gNg⁻¹ = N, without changing the set of subgroups that qualify. Equally, normality is equivalent to the left and right cosets of N in G coinciding: for every g, the set gN equals the set Ng. A more structural reformulation says that N is a union of entire conjugacy classes of G, meaning it is built from complete orbits under the conjugation action. Another perspective frames normality as preservation under all inner automorphisms of G. Finally, and perhaps most powerfully, N is normal if and only if it appears as the kernel of some group homomorphism whose domain is G. These parallel characterizations reveal that normality is not a single property but a convergence of many structural demands.

Quotient Groups and the Classification of Homomorphisms

The practical importance of normal subgroups becomes unmistakable when one asks which subgroups can be factored out of a group. The answer is exclusive: normal subgroups, and only normal subgroups, permit the construction of a quotient group. When N is normal in G, the collection of left cosets of N inherits a well-defined multiplication rule—multiplying the coset gN by the coset hN produces the coset (gh)N—thereby forming a new group denoted G/N. This works because normality guarantees that the equivalence relation of belonging to the same left coset is preserved under the group operation, so the product of two cosets does not depend on which representatives are chosen. Beyond quotient construction, normal subgroups play a dual role: they are exactly the kernels of all group homomorphisms that have G as their domain. This means the family of normal subgroups of G provides a complete internal classification of every homomorphism emanating from G, linking the internal structure of G to the maps it can send to other groups.

Galois and the Birth of a Central Idea

The recognition that normal subgroups matter was neither immediate nor obvious; it required a genuine leap of insight. Évariste Galois is credited as the first mathematician to appreciate the significance of the existence of normal subgroups within a group. In the broader landscape of abstract algebra, this recognition proved foundational. The concept of normality sits at the intersection of several major themes: it governs which substructures can be quotiented away, it identifies the kernels that classify homomorphisms, and it connects to the symmetry of conjugacy classes and inner automorphisms. Because every one of these roles depends on the single invariance condition gng⁻¹ ∈ N, the normal subgroup became a unifying thread through the structure theory of groups. Galois's early identification of this thread gave later algebraists a tool that remains indispensable: a way to decompose a group into simpler pieces, to understand its maps to other groups, and to recognize the internal symmetries that make a subgroup fit its parent.