# Internal direct product

This article describes a product notion for groups. See other related product notions for groups.

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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:

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:

1. Each  is a normal subgroup of 
2. The s generate 
3. 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:

1. Every element of  commutes with every element of  for 
2. 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.