# direct products of groups

Let ${({G}_{i})}_{i\in I}$ be a family of groups.

The *unrestricted direct product*
(or *complete direct product*, or *Cartesian product*)
${\prod}_{i\in I}{G}_{i}$
is the Cartesian product (http://planetmath.org/GeneralizedCartesianProduct)
${\prod}_{i\in I}{G}_{i}$ with multiplication^{} defined pointwise,
that is, for all $f,g\in {\prod}_{i\in I}{G}_{i}$ and all $i\in I$
we have $(fg)(i)=f(i)g(i)$.
It is easily verified that this multiplication
makes the Cartesian product into a group.
This construction is in fact the categorical direct product (http://planetmath.org/CategoricalDirectProduct) in the category of groups.

The *restricted direct product ^{}* ${\oplus}_{i\in I}{G}_{i}$ is the subgroup

^{}of ${\prod}_{i\in I}{G}_{i}$ consisting of all those elements with finite support. That is,

$$\underset{i\in I}{\oplus}{G}_{i}=\left\{f\in \prod _{i\in I}{G}_{i}\right|f(i)=1\text{for all but finitely many}i\in I\}.$$ |

The restricted direct product is also called the *direct sum ^{}*, although this usage is usually reserved for the case where all the ${G}_{i}$ are abelian

^{}(see direct sum of modules (http://planetmath.org/DirectSum) and categorical direct sum (http://planetmath.org/CategoricalDirectSum)).

The unqualified term *direct product ^{}* can refer either to the unrestricted direct product or to the restricted direct product, depending on the author.
Note that if $I$ is finite then the unrestricted direct product and the restricted direct product are in fact the same.
The direct product of two groups $G$ and $H$ is usually written $G\times H$,
or sometimes $G\oplus H$ (or $G\coprod H$) if $G$ and $H$ are both abelian.

Title | direct products of groups |

Canonical name | DirectProductsOfGroups |

Date of creation | 2013-03-22 14:52:54 |

Last modified on | 2013-03-22 14:52:54 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 17 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20A99 |

Related topic | SubdirectProductOfGroups |

Related topic | DirectProduct2 |

Defines | direct product |

Defines | unrestricted direct product |

Defines | complete direct product |

Defines | restricted direct product |

Defines | direct sum |

Defines | direct product of groups |

Defines | unrestricted direct product of groups |

Defines | restricted direct product of groups |

Defines | direct sum of groups |

Defines | Cartesian product of g |