Direct Product Article Index for
Direct 		Website Links For
Direct 		 

Information About

Direct Product
  CATEGORIES ABOUT DIRECT PRODUCT
   
abstract algebra
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



already known, giving a new one. Examples are the product of sets (see Cartesian Product ), groups (described below), the Product Of Rings and of other Algebraic Structures . The Product Of Topological Spaces is another instance.


GROUP DIRECT PRODUCT

In Group Theory one can define the direct product of two
  • ) and (''H'', o), denoted by ''G'' × ''H''. For Abelian Group s which are written additively, it is also called the Direct Sum , denoted by G \oplus H.


It is defined as follows:
  • as Set of the elements of the new group, take the '' Cartesian Product '' of the sets of elements of ''G'' and ''H'', that is {(''g'', ''h''): ''g'' in ''G'', ''h'' in ''H''};

  • on these elements put an operation, defined elementwise:
    (''g'', ''h'') × (''g' '', ''h' '') = (''g'' --- ''g' '', ''h'' o ''h' '')

  • may be the same as o.)


This construction gives a new group. It has a Normal Subgroup
Isomorphic to ''G'' (given by the elements of the form (''g'', 1)),
and one isomorphic to ''H'' (comprising the elements (1, ''h'')).

The reverse also holds, there is the following recognition theorem: If a group ''K'' contains two normal subgroups ''G'' and ''H'', such that ''K''= ''GH'' and the intersection of ''G'' and ''H'' contains only the identity, then ''K'' = ''G'' x ''H''. A relaxation of these conditions gives the Semidirect Product .

As an example, take as ''G'' and ''H'' two copies of the unique (up to
  • (''a'',1) = (1---''a'', ''b''---1) = (''a'',''b''), and (1,''b'')---(1,''b'') = (1,''b''2) = (1,1).


With a direct product, we get some natural Group Homomorphisms for free: the projection maps
:\pi_1 \colon G imes H o G\quad \mathrm{by} \quad \pi_1(g, h) = g,
:\pi_2 \colon G imes H o H\quad \mathrm{by} \quad \pi_2(g, h) = h
called the coordinate functions.

Also, every homomorphism ''f'' on the direct product is totally determined by its component functions
f_i = \pi_i \circ f.

  • ), and any integer ''n'' ≥ 0, multiple application of the direct product gives the group of all ''n''- Tuple s ''G''''n'' (for ''n''=0 the trivial group). Examples:

  • Z''n''

  • R''n'' (with additional Vector Space structure this is called Euclidean Space , see below)



VECTOR SPACE DIRECT PRODUCT

The direct product for Vector Spaces (not to be confused with the Tensor Product ) is very similar to the one defined for groups above, using the Cartesian Product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components. Starting from R we get Euclidean Space R''n'', the prototypical example of a real ''n''-dimensional vector space. The vector space direct product of R''m'' and R''n'' is R''m'' + ''n''.

Note that a direct product for a finite index \prod_{i=1}^n X_i is identical to the Direct Sum \bigoplus_{i=1}^n X_i . The direct sum and direct product differ only for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries.


TOPOLOGICAL SPACE DIRECT PRODUCT

The direct product for a collection of Topological Spaces ''Xi'' for ''i'' in ''I'', some index set, once again makes use of the cartesian product

:\prod_{i \in I} X_i

Defining the of open sets to be the collection of all cartesian products of open subsets from each factor:

  :<math>\mathcal B \left\{ \prod_{i \in I} U_i\ \ (\exists j_1,\ldots,j_n)(U_{j_i}\ \mathrm{open\ in}\ X_{j_i})\ \mathrm{and}\ ( orall i