Given two groups 
and 
, there are several ways to form a new group. The simplest
is the direct product, denoted 
. As a set, the group direct product is the Cartesian
product of ordered pairs 
, and the group operation is componentwise, so
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For example, 
is isomorphic to 
under vector addition. In a similar fashion, one
can take the direct product of any number of groups by taking the Cartesian product
and operating componentwise.
Note that 
is isomorphic to the subgroup
of elements 
where 
is the identity element in 
. Similarly, 
can be realized as a subgroup.
The intersection of these two subgroups is the identity 
, and the two subgroups are normal.
Like the ring direct product, the group direct product has the universal property that if
any group 
has a homomorphism to 
and a homomorphism to 
, then these homomorphisms factor through 
in a unique way.
If one has group representations 
of 
and 
of 
, then there is a representation 
sometimes called the external
tensor product, given by the tensor product 
. In this case, the group
character satisfies
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