A normal subgroup is a subgroup of a group that is invariant under conjugation by any element in the group. In other words, for any element in the normal subgroup, if we multiply it by any element in the original group and then multiply it by the inverse of that element, the result will still be in the normal subgroup.
A normal subgroup is important because it allows us to define a quotient group, which is a group formed by the cosets of the normal subgroup. This allows us to study the structure of the original group in a more manageable way.
The kernel of a homomorphism is the set of elements in the domain that are mapped to the identity element in the codomain. Since the kernel is a normal subgroup, it allows us to define a quotient group and study the structure of the original group in a more manageable way.
A kernel is related to a homomorphism in that it is the set of elements in the domain that are mapped to the identity element in the codomain. This allows us to study the structure of the original group and better understand the properties of the homomorphism.
No, not every normal subgroup can be represented as a kernel. However, every kernel is a normal subgroup. This means that there are some normal subgroups that cannot be represented as a kernel, but every normal subgroup that can be represented as a kernel is also a normal subgroup.