What is the definition of linearly independent subset of an abelian group?
I know the concept of this, but i don't know how to define this term precisely.
Below is what i tried to formulate:
Let $G$ be an abelian group with the identity $e$.
Let $S\subset G$
Define $\text{supp}(f)\triangleq \{x\in G: f(x)\neq 0\}$ for any function $f:G\rightarrow \mathbb{Z}$.
Then $S$ is linearly independent iff $\forall f\in \mathbb{Z}^S, [\text{ supp}(f) \text{ is finite and } \sum_{x\in\text{supp}(f)}f(x)x=e] \Rightarrow [\forall x\in S, f(x)=0$]
Is this a right formulation?
If not, what would be the precise definition of linearly independence of abelian group?
Linear independence in an Abelian group should be an example of linear independence in a module. Each Abelian Group is a module over the integers, since we can multiply an element by a positive integer by adding it to itself that many times, and by a negative integer by adding the inverse to itself that many times. The definition of linear independence for a set $S \subset G$ is that if we have any $g_1, g_2, \ldots , g_n \in S$ the only integer coefficients $d_1, d_2, \ldots, d_n$ such that $d_1 g_1 + \cdots +d_n g_n = 0$ are all $d_i = 0$. This is more or less what you have. So you are correct.