Direct factor versus normal
This survey article compares, and contrasts, the following subgroup properties: direct factor versus normal subgroup
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This article is about the relation, the similarity and contrast, between the well-known subgroup property of normality, and the important subgroup property of being a direct factor -- a factor in a direct product.
Further information: normal subgroup
A subgroup in a group is termed normal (in symbols, ) if it satisfies the following equivalent conditions:
- is the kernel of a homomorphism from . In particular, there is a quotient group .
- Every inner automorphism of leaves invariant. In symbols, for every .
Further information: direct factor
A subgroup of a group is termed a direct factor if it is normal, and there is another subgroup of such that is normal, and is trivial.
Such a will be termed a normal complement to .
Direct factor implies normal
Further information: direct factor implies normal
Normal not implies direct factor
Further information: normal not implies direct factor
Corresponding notions of simplicity
Further information: simple group
Directly indecomposable group
Further information: directly indecomposable group
Similar role in the factorization sense
Normal subgroups as a tool for factorization
Just as we factorize natural numbers as products of smaller natural numbers, we want to factorize groups in terms of smaller subgroups. One such approach is using normal subgroups. If , then we can intuitively think of as being factored into two pieces and .
This analogy, though, fails in the following senses:
- The role played by and in the factorization is not symmetric, and in this sense, the factorization does not commute. While is a subgroup, is a quotient group. There are situations where we can identify with a subgroup of that intersects trivially (this is the situation of an internal semidirect product) but this is not always true.
- The structure of is not determined by the structure of and that of . In other words, knowing and upto isomorphism doesn't reveal upto isomorphism. There is additional pasting information that describes how and interact. This is in sharp contrast to the case of numbers, where we can multiply the factors together to get back the original number.
One can use factorization in the sense of normal subgroups to get the notion of composition series: a subnormal series with the property that all successive quotient groups are simple. A uniqueness theorem of sorts holds for all groups admitting a composition series of finite length: any two composition series have the same collection of isomorphism types of simple groups, with the same multiplicity of each isomorphism type.
Direct factors as tools of factorization
Factorization of natural numbers may seem to bear closer resemblance to the notion of direct factors in a group. Namely, if is a direct factor, with a normal complement , we may think of as factored into and . is isomorphic to the quotient group , and is isomorphic to the quotient group , so this is a special case of the general factorization in terms of normal subgroups.
This approach doesn't have the two failings mentioned above for normal subgroups:
- The role played by and is symmetric. Thus, the factorization is commutative.
- is completely determined by and . In fact, it is isomorphic to the external direct product .
On the other hand, knowing does not completely determine as a subgroup. Although any normal complement to must be isomorphic to , there could be many different normal complements to .
Direct factors can be used as a tool of factorization. Every finite group is expressible as a direct product of directly indecomposable groups, and the Remak-Schmidt theorem states that the isomorphism types and multiplicities of these pieces are independent of the choice of decomposition.