Direct factor versus normal
This survey article compares, and contrasts, the following subgroup properties: direct factor versus normal subgroup
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Introduction
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.
Definitions
Normal subgroup
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
.
Direct factor
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.
(The three conditions are equivalent to saying that is the internal direct product of
with
. Thus,
is isomorphic to the external direct product
).
Such a will be termed a normal complement to
.
Implication relations
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
Simple group
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.