Question:Normal subgroup direct product

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This question is about normal subgroupdirect factor, internal direct product, and external direct product| See more questions about normal subgroup | See more questions about direct factor
This question has type relation between two concepts

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:

Characteristic subgroup of direct factor, Complemented central factor, Complemented normal subgroup, Complemented transitively normal subgroup, Conjugacy-closed normal subgroup, Direct factor over central subgroup, Endomorphism kernel, Intermediately endomorphism kernel, Join of finitely many direct factors, Join-transitively central factor, Locally inner automorphism-balanced subgroup, Normal AEP-subgroup, Normal subgroup having a 1-closed transversal, Normal subgroup in which every subgroup characteristic in the whole group is characteristic, Normal subgroup whose focal subgroup equals its derived subgroup, Powering-invariant normal subgroup, Quotient-powering-invariant subgroup, Right-quotient-transitively central factor, SCAB-subgroup, Transitively normal subgroup... further results|FULL LIST, MORE INFO