Normal Subgroup

Definition

A normal subgroup is a subgroup that is invariant under conjugation by any element of the group. This means that for every element in the normal subgroup and every element in the larger group, the result of conjugating the subgroup element by the group element will still be in the subgroup. Normal subgroups are significant because they allow for the construction of quotient groups, which are fundamental in understanding group structures and their symmetries.

5 Must Know Facts For Your Next Test

1. A subgroup N of a group G is normal if for all g in G, the conjugate gNg^{-1} is still in N.
2. Normal subgroups play a critical role in defining homomorphisms between groups, allowing one to map groups to simpler structures.
3. The kernel of a homomorphism is always a normal subgroup of the original group.
4. Every group has at least two normal subgroups: the trivial subgroup and the group itself.
5. The intersection of two normal subgroups is also a normal subgroup.

Review Questions

• How does the property of being a normal subgroup relate to conjugation within a group?
  • A normal subgroup is defined by its behavior under conjugation, meaning that if you take any element from the group and conjugate an element from the normal subgroup, the result will still belong to that normal subgroup. This property ensures that the structure of the group remains consistent even when elements are combined in this way. Essentially, this invariance under conjugation is what differentiates normal subgroups from other types of subgroups.
• Discuss how normal subgroups facilitate the formation of quotient groups and their importance in algebraic structures.
  • Normal subgroups allow for the construction of quotient groups by enabling the grouping of elements into cosets. When we have a normal subgroup N of a group G, we can create cosets like gN for each element g in G. The set of these cosets forms a new group called the quotient group G/N. This process is crucial because it simplifies complex group structures, making it easier to study their properties and relationships with other groups.
• Evaluate the role of normal subgroups in understanding homomorphisms and their impact on algebraic structures.
  • Normal subgroups are integral to understanding homomorphisms because they form the kernel of such mappings. The kernel is always a normal subgroup since it consists of elements that map to the identity element in another group. By analyzing these kernels, we gain insights into how groups relate to each other through homomorphisms. This relationship showcases how different algebraic structures can be interconnected, allowing us to explore deeper properties and classifications within group theory.

Related terms

Conjugation: The process of taking an element from a group and multiplying it by another element and its inverse, typically expressed as $gAg^{-1}$ for a subgroup A and element g.

Quotient Group: A new group formed by partitioning a group into cosets of a normal subgroup, denoted as G/N, where G is the original group and N is the normal subgroup.

Simple Group: A nontrivial group that does not have any normal subgroups other than the trivial group and itself, indicating it cannot be broken down into smaller groups.