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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?

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    $\begingroup$ Not sure what the question is. Amongst a collection of would-be generators we can ask if one of them can be written as a word in the others. In that way, you might hope to reduce a given set to a minimal subset. Of course, you can ask if there might be a smaller set still..made from different generators, but questions like that tend to be profoundly difficult. $\endgroup$ – lulu May 30 '16 at 19:47
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    $\begingroup$ Sure. Knowing that a group is finitely generated tells us a lot. And you are definitely correct that the size of the generating set is not a well defined property of the group. $\endgroup$ – lulu May 30 '16 at 20:10
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    $\begingroup$ Abstractly presented groups can be pretty gruesome. For example, a subgroup of a finitely generated group need not be finitely generated. See this for the standard counterexample. $\endgroup$ – lulu May 30 '16 at 20:12
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    $\begingroup$ Yes, that's a perfectly good example. $\endgroup$ – lulu May 30 '16 at 20:16
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    $\begingroup$ Not entirely sure what point you are making. Yes...a free group on $n$ letters is not isomorphic to a free group on $m$ letters for $n\neq m$. is that what you mean? But a finite symmetric group is generated by all its transpositions...or by one transposition and an $n$-cycle. So... $\endgroup$ – lulu May 31 '16 at 19:24
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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 $

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    $\begingroup$ But the number of elements of a free system that generat a finitaly group is not an invariant for this group: exemple the group $ S_n$ finite with two free system an of cardinal 2 and other n-1, $\endgroup$ – m.idaya May 30 '16 at 20:20
  • $\begingroup$ Are you sure the two systems are free ? $\endgroup$ – palio May 31 '16 at 19:42
  • $\begingroup$ each of the above systems is a minimal generator system, ie all strictly subset of each of these systems does not generate the entir group.I specify systems $A_1=\{(12),(123\cdot\cdot\cdot n)\}$ and $A_2=\{(i\; i+1), 1\leq i\leq n-1)\}$ $\endgroup$ – m.idaya May 31 '16 at 22:05

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