A normal subgroup is a subgroup that remains unchanged under conjugation by elements of its parent group. In other words, for any element in the parent group, if you apply it to a normal subgroup, the result will still be within the normal subgroup.
A normal subgroup is a specific type of subgroup that satisfies the condition of remaining unchanged under conjugation. Regular subgroups do not necessarily have this property.
Normal subgroups are important because they allow for the creation of quotient groups, which can help simplify the study of larger groups. They also have several applications in algebra and geometry.
The trivial subgroup and the parent group itself are always normal subgroups. Other examples include the center of a group, the commutator subgroup, and the kernel of a group homomorphism.
Normal subgroups have applications in many areas, such as cryptography, physics, and chemistry. For example, they are used in coding theory and in the study of particle symmetries.