The group direct product 
of two groups 
and 
is the Cartesian product
of ordered pairs 
with the componentwise group
operation
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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 group elements 
where 
is the identity element
in 
.
Similarly, 
can be realized as a subgroup. The intersection
of these two subgroups is the singleton
set 
containing the identity element, 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 group homomorphism
to 
and a group homomorphism to 
, then these group homomorphisms
factor through 
in a unique way.
If one has group representations 
of 
and 
of 
, then there is a group
representation 
sometimes called the external tensor product,
given by the vector space tensor product 
. In this case, the group
character satisfies
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