There are 3 questions about minimal generating sets of groups when I am trying to prove that every subgroup of a free group is free, using Cayley graphs.
I firstly give my definition of it. For a group and a generating set $S$ of it, if any proper subset of $S$ is not a generating set, then call $S$ minimal.
Then the questions are as below:
Do generating sets of an arbitrary group always exist? Generating sets always exists because groups themselves are their own generating sets, but they're not minimal.
Is a free group freely generated by any minimal generating set?
For a subset $S$ of a group, can we find a minimal generating set containing $S$?
These questions seem to be obvious when I first came up with, but I didn't find a strict proof.
Some groups do not have minimal generating sets, as mentioned in the comments. For example consider $\mathbf{Q}$. In fact you can remove any element from any generating set of $\mathbf{Q}$ and the remainder must still generate. This answers questions 1 and 3 negatively.
Question 2: $F_1 = \mathbf Z$ has a minimal generating set $\{2,3\}$, but clearly it is not freely generated by $2$ and $3$.
What is true is that $F_n$ is freely generated by any generating set of size $n$. See here or here.