- #1
raj123
- 16
- 0
Normal subgroups??
Normal subgroups?
Normal subgroups?
Last edited:
A normal subgroup is a subset of a group that is closed under the group operation and also satisfies a certain condition called normality. This condition means that the normal subgroup is invariant under conjugation by elements of the larger group.
A normal subgroup is a special type of subgroup that has the property of normality, while a subgroup does not necessarily have this property. This means that a normal subgroup is preserved under conjugation by elements of the larger group, whereas a subgroup may not be.
Normal subgroups are important in the study of groups because they allow us to define quotient groups, which are groups formed by taking the larger group and "modding out" the normal subgroup. This allows us to study the structure of the original group in a simpler way.
There are a few ways to determine if a subgroup is normal. One way is to check if every element of the subgroup is invariant under conjugation by elements of the larger group. Another way is to see if the subgroup is the kernel of a homomorphism from the larger group to another group.
Yes, a group can have multiple normal subgroups. In fact, the trivial subgroup (containing only the identity element) and the entire group itself are always normal subgroups of any group. It is also possible for a group to have only one normal subgroup or no normal subgroups at all.