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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?

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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.

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