I'm learning about the concept of a free abelian group. First question: nowhere it is stated that these groups cannot be finite, but the definition seems to imply it. Is this true?
Second question: an isomorphism to a free abelian group $A = \left< S \right> = \left< \{s_i\}_{i=1}^n \right>$ is given as
\begin{eqnarray*} &ℤ^n &\tilde\longrightarrow A \\ &(c_i)_{i=1}^n &\mapsto \sum_{s ∈ S} c_ss \end{eqnarray*}
Then a claim is made that I don't understand:
"The induced isomorphism $A/2A \cong (ℤ/2ℤ)^n$ now shows that $A/2A$ is of order $2^n$, from which we conclude that the rank of $A$ does not depend on the choice of a basis for $A$"
What is this induced isomorphism? Is it induced by the (first) Isomorphism Theorem? What exactly is $A/2A$, and why are we looking at it? How is the final conclusion drawn?
The trivial group is free abelian (of rank $0$), and finite. But every nontrivial free abelian group is infinite: given an element $s$ of the free basis, you can map $s$ to the generator of the infinite cyclic group $C_{\infty}$ and ever other element of the basis to the identity. The universal property gives you a homomorphism from the free abelian group to this infinite group, so the free abelian group has an infinite quotient, hence is itself infinite.
Under the original map, the subgroup $2A$ corresponds to the subgroup $(2\mathbb{Z})^n$. If you have a morphism $f\colon G\to K$, a normal subgroup $M\triangleleft K$, and a normal subgroup $N\triangleleft G$ such that $f(N)\subseteq M$, then you get an “induced map” $(G/N)\to (K/M)$ given by $gN\mapsto f(g)M$. In the case at hand, $f$ is an isomorphism, $2A$ corresponds to $(2\mathbb{Z})^n$, so you actually get an isomorphism between the quotients.
$A/2A$ is just the quotient of $A$ modulo the “even” elements. It gives you an abelian group such that $2a=0$ for all $a$; this makes it into a vector space over the field of $2$ elements, and we know lots of things about such vector spaces (in particular, that they have a well-defined dimension). So we are looking at it to leverage that knowledge into showing that the size of a basis of a free abelian group is unique.