Normal Subgroup

Definition

A normal subgroup is a subgroup that is invariant under conjugation by members of the group, meaning that for every element in the group, when you conjugate any element of the subgroup by that group element, the result stays within the subgroup. This property is crucial because it allows for the construction of quotient groups, which are fundamental in understanding group structure and symmetry. Additionally, normal subgroups play a key role in the context of Galois theory, where they relate to the decomposition and inertia groups associated with field extensions.

5 Must Know Facts For Your Next Test

1. A subgroup H of a group G is normal if for all g in G and h in H, the element gHg^{-1} is contained in H.
2. Normal subgroups are essential for constructing quotient groups, which simplify the study of group structure.
3. Every normal subgroup is automatically a characteristic subgroup, meaning it is invariant under all automorphisms of the parent group.
4. In Galois theory, normal subgroups correspond to certain types of field extensions known as normal extensions, which have nice algebraic properties.
5. The intersection of any two normal subgroups is also a normal subgroup, making the collection of normal subgroups behave well under intersection.

Review Questions

• How does the property of being a normal subgroup facilitate the construction of quotient groups?
  • Being a normal subgroup allows for the formation of quotient groups because it ensures that the left and right cosets of the subgroup coincide. This means that when we partition the original group by this normal subgroup, we can define a well-defined group operation on these cosets. Without normality, the cosets might not behave consistently under multiplication, preventing us from forming a legitimate group structure.
• Discuss how normal subgroups are utilized in Galois theory and their impact on understanding field extensions.
  • In Galois theory, normal subgroups correspond to normal extensions of fields. These are extensions where every irreducible polynomial with roots in the extension splits completely into linear factors. The structure of Galois groups reflects this relationship; if a subgroup is normal, it gives rise to a quotient that captures important information about the symmetries of these field extensions. This connection helps us analyze how various polynomials behave and allows us to classify extensions effectively.
• Evaluate the significance of normal subgroups in both group theory and algebraic number theory contexts.
  • Normal subgroups are significant because they serve as bridges between group theory and algebraic structures in number theory. In group theory, they enable us to create quotient groups which provide insights into the overall group's structure. In algebraic number theory, normal subgroups help us analyze symmetries and properties within Galois groups related to polynomial roots and field extensions. This dual importance illustrates how central normal subgroups are to understanding both abstract algebraic concepts and their applications in number theory.

Related terms

Conjugate: The result of taking an element 'g' from a group and an element 'h' from a subgroup, then forming 'g * h * g^{-1}', where '*' denotes the group operation.

Quotient Group: A group formed by partitioning a group into cosets of a normal subgroup, allowing for a new structure that reflects properties of the original group.

Galois Group: The group of field automorphisms that fix a base field and describe symmetries in the roots of polynomials, closely linked to normal subgroups when examining field extensions.