Internal direct product
This article describes a product notion for groups. See other related product notions for groups.
Contents |
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition
Definition with symbols (for two subgroups)
A group is termed the internal direct product of two subgroups
and
if both the following conditions are satisfied:
-
and
are both normal subgroups
-
and
are permutable complements, that is,
is trivial and the product of subgroups
.
Equivalently is the internal direct product of
and
if both the following conditions are satisfied:
- Every element of
commutes with every element of
. In other words,
is contained in the centralizer of
.
-
and
are lattice complements, that is, they intersect trivially and together they generate
, i.e., the join of subgroups
is equal to
.
The two subgroups and
are termed direct factors of
.
Definition with symbols (for arbitrary family of subgroups)
A group is termed the internal direct product of subgroups
, if the following three conditions are satisfied:
- Each
is a normal subgroup of
- The
s generate
- Each
intersects trivially the subgroup generated by the other
s. Equivalently, if
where
with all
distinct, then each
.
Equivalently, is the internal direct product of the
s if the following two conditions are satisfied:
- Every element of
commutes with every element of
for
- Each
is a lattice complement to the subgroup generated by the remaining
s
Equivalence of definitions
Further information: equivalence of definitions of internal direct product
Equivalence with the external direct product
Further information: equivalence of internal and external direct product
It can be proved that if is an internal direct product of subgroups
and
, then
is isomorphic to the external direct product
×
via the isomorphism that sends a pair
from
to the product
in
. Conversely, given an external direct product
, we can find subgroups isomorphic to
and
in the external direct product such that it is the internal direct product of those subgroups.
For infinite collections of subgroups, the internal direct product does not coincide with the external direct product -- instead, it coincides with the notion of restricted external direct product.
Relation with other properties
Weaker product notions
- Semidirect product where only one of the subgroups is assumed to be normal
- Exact factorization where neither subgroup is assumed to be normal
- Group extension where there is a normal subgroup and a quotient (the quotient may not occur as a subgroup)
- Regular product
- Verbal product
- Reduced direct product
- Subdirect product