Editing Direct product (section)

== Direct product of groups ==
{{main|Direct product of groups|Direct sum}}
In [[group (mathematics)|group theory]], define the direct product of two groups <math>(G, \circ)</math> and <math>(H, \cdot),</math> can be denoted by <math>G \times H.</math> For [[abelian group|abelian groups]] that are written additively, it may also be called the [[direct sum of groups|direct sum of two groups]], denoted by <math>G \oplus H.</math>

It is defined as follows:
* the [[Set (mathematics)|set]] of the elements of the new group is the ''Cartesian product'' of the sets of elements of <math>G \text{ and } H,</math> that is <math>\{(g, h) : g \in G, h \in H\};</math>
* on these elements put an operation, defined element-wise: <math display="block">(g, h) \times \left(g', h'\right) = \left(g \circ g', h \cdot h'\right)</math>
Note that <math>(G, \circ)</math> may be the same as <math>(H, \cdot).</math>

The construction gives a new group, which has a [[normal subgroup]] that is isomorphic to <math>G</math> (given by the elements of the form <math>(g, 1)</math>) and one that is isomorphic to <math>H</math> (comprising the elements <math>(1, h)</math>).

The reverse also holds in the recognition theorem. If a group <math>K</math> contains two normal subgroups <math>G \text{ and } H,</math> such that <math>K = GH</math> and the intersection of <math>G \text{ and } H</math> contains only the identity, <math>K</math> is isomorphic to <math>G \times H.</math> A relaxation of those conditions by requiring only one subgroup to be normal gives the [[semidirect product]].

For example, <math>G \text{ and } H</math> are taken as two copies of the unique (up to isomorphisms) group of order 2, <math>C^2:</math> say <math>\{1, a\} \text{ and } \{1, b\}.</math> Then, <math>C_2 \times C_2 = \{(1,1), (1,b), (a,1), (a,b)\},</math> with the operation element by element. For instance, <math>(1,b)^* (a,1) = \left(1^* a, b^* 1\right) = (a, b),</math> and<math>(1,b)^* (1, b) = \left(1, b^2\right) = (1, 1).</math>

With a direct product, some natural [[group homomorphisms]] are obtained for free: the projection maps defined by
<math display=block>\begin{align}
  \pi_1: G \times H \to G, \ \ \pi_1(g, h) &= g \\
  \pi_2: G \times H \to H, \ \ \pi_2(g, h) &= h
\end{align}</math>
are called the '''coordinate functions'''.

Also, every homomorphism <math>f</math> to the direct product is totally determined by its component functions <math>f_i = \pi_i \circ f.</math>

For any group <math>(G, \circ)</math> and any integer <math>n \geq 0,</math> repeated application of the direct product gives the group of all <math>n</math>-[[tuples]]  <math>G^n</math> (for <math>n = 0,</math> that is the [[trivial group]]); for example, <math>\Z^n</math> and <math>\R^n.</math>