When we say a group $G$ is finitely generated we mean it can be generated by a finite number of elements but this does not exclude the possibility of being generated by an infinite number of elements of $G$. So saying $G$ is finitely generated is not saying that it is only finitely generated but rather it is to insist on the fact that it is possible to be generated with a finite number of elements, something that is important in the study of the group and also it is something that is not satisfied by all the groups, for example the group of rationals $(\mathbb Q,+)$ cannot be generated by a finite number of rational numbers, but can for example be generated by the infinite subset of rationals $\{\frac{1}{n}\;|n\in \mathbb N^*\;\}$. Also when the group is finitely generated, the number of elements is not a characteristic of the group, for example $\mathbb Z$ can be generated by one element $\{1\}$ or by two coprime integers, so we just need to know that the group is finitely generated without giving much importance to the number of the generating elements, but sometimes it seems to me that we look at the minimum number of elements that can generate the group: for example in the group of integers we give more importance to the fact that it can be generated by only one element and call it a cyclic group for that. Also in the case of an $F$-vector space we look at linear dependence between these generators induced from multiplication with scalars from the field $F$ but in a group what sort of dependence we are looking for?
[Math] Finitely generated vs infinitely generated group
group-theory
Best Answer
in group theory, I think better to say, a subset S of group G is called free if any strict subset A of S we have $\langle A\rangle \varsubsetneq \langle S\rangle $