# Finitely Generated Abelian Groups

The Fundamental Theorem of Finitely Generated Abelian Groups states:

Let $$G$$ be a finitely generated abelian group. Then it decomposes as follows: $$G \cong Z^r \times Z_{n_1} \times Z_{n_2} \times \cdot \cdot \cdot \times Z_{n_s},$$ for some integers $$r,n_1,n_2, \cdot \cdot \cdot,n_s$$ satisfying the following conditions:

• $$r≥0$$ and $$n_i≥2$$ for all $$i$$, and

• $$n_{i+1}|n_i$$ for $$1≤i≤s−1$$.

The decomposition of $$G$$ satisfying these conditions is unique.

If $$r=0$$, I know what these groups will look like. They are finite, and elements in the group are tuples where the $$i$$th element is an element of the $$ith$$ cylic group.

But it $$r>0$$, then the group is infinite. This means that there are some indices of the tuple where we know that the element will have one of a finite option of values. But, for $$r$$ elements of the tuple, we don't know anything about the element, because there are infinite choices.

I was wondering if this was a good way to think about these groups? Do we treat elements in these infinite groups like $$(s+r) -$$ tuples?

For example, you can visualize $$\mathbb Z^2$$ as the integer lattice in the plane, meaning the points $$(m,n)$$ in the coordinate plane $$\mathbb R^2$$ with integer coordinates $$m$$ and $$n$$. Furthermore, you can visualize the group operation on $$\mathbb Z^2$$ as vector addition in the plane.
The same visualization continues into three dimensions: $$\mathbb Z^3$$ is the integer lattice in 3-space, meaning the points $$(l,m,n)$$ of coordinate 3-space $$\mathbb R^3$$ with integer coordinates $$l$$, $$m$$ and $$n$$. Again, the group operation is just vector addition.
Our minds are not so good at visualizing higher dimensions, nonetheless we have a mathematical theory of coordinate $$n$$-space $$\mathbb R^n$$, and $$\mathbb Z^n$$ is just the integer lattice in $$\mathbb R^n$$, meaning the points with integer coordinates.