Question:Normal subgroup direct product

Q: '''I learned something about normal subgroup and also saw the definitions of internal direct product and external direct product. Are normal subgroups basically the things which arise in direct products?'''

A: The external direct product is a general algebraic construction that applies specifically to groups. It is related to the internal direct product, where a group is viewed as being like a direct product of subgroups of it. These notions are equivalent.

A subgroup of a group which arises as one of the factors in an internal direct product decomposition is termed a direct factor. Any direct factor is normal, but the converse is not true: normal not implies direct factor. Basically, a direct factor is a normal subgroup that has a normal complement, i.e., there must exist another normal subgroup intersecting it trivially and generating the whole group along with it. The condition of being a direct factor is considerably stronger than being normal. Most normal subgroups are not direct factors.

There are many intermediate notions between normal subgroup and direct factor. A complemented normal subgroup is a normal subgroup possessing a (not necessarily normal) complement --such a complement is called a retract. An endomorphism kernel is a normal subgroup that arises as the kernel of an endomorphism. A fuller list of intermediate properties between normal subgroup and direct factor is below: