Every finitely generated Abelian group can be decomposed by taking this operation over a series of cyclic groups. Taking this operation on a set of groups creates a new group with a pair of homeomorphisms to the original groups called the coordinate functions. Taking it on a series of compact topological spaces returns a compact topological space according to Tychonoff's theorem while taking it on a pair of sets returns a set of ordered pairs when taking the Cartesian form of this operation. A ring whose nonzero elements form an Abelian group under this operation is called a field. This operation doesn't have inverses for the integers, but do
Every finitely generated Abelian group can be decomposed by taking this operation over a series of cyclic groups. Taking this operation on a set of groups creates a new group with a pair of homeomorphisms to the original groups called the coordinate functions. Taking it on a series of compact topological spaces returns a compact topological space according to Tychonoff's theorem while taking it on a pair of sets returns a set of ordered pairs when taking the Cartesian form of this operation. A ring whose nonzero elements form an Abelian group under this operation is called a field. This operation doesn't have inverses for the integers, but does for the rational numbers. For 10 points, name this operation that can be viewed as repeated addition
Every finitely generated Abelian group can be decomposed by taking this operation over a series of cyclic groups.
product [or multiplying; or Cartesian product; or direct product]