In abstract algebra
, the free abelian group
on a set X
may be constructed as the abelian group
of functions on X
, taking integer values that are almost all
zero. One can verify directly that this has the appropriate universal property
in relation to arbitrary functions on X
with values in some abelian group A
: namely unique extension to a homomorphism of the free group.
When X is finite of cardinality n the free abelian group on X is the same up to isomorphism as the product of n copies of the infinite cyclic group. This breaks down for infinite X, though.
This construction is a special case of the construction of free modules[?].
All Wikipedia text
is available under the
terms of the GNU Free Documentation License