Internal direct product
This article describes a product notion for groups. See other related product notions for groups.
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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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