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Is that's all? Thank you. :-)

A group $H$ is called finitely generated if there is a finite set $A$ such that $H = \left \langle A \right \rangle$ . Prove that every finite group is finitely generated.

Because $\forall a \in A, a\in H$. Hence, if $A$ is infinite, $H$ is infinite. Hence every finite group is finitely generated.

Tumbleweed
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    You forgot to tell us that $H$ is a group and $A\subseteq H$ is a generating set. See http://math.stackexchange.com/q/66256/ – Jonas Meyer Sep 04 '13 at 04:50
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    The group elements are a set of generators. There are finitely many. Note that there are many infinite finitely generated groups. So if your post says that if a group is infinite, it is not finitely generated, then that part is not correct. – André Nicolas Sep 04 '13 at 04:52
  • Thanks @AndréNicolas - so as what I proved is correct, right? – Tumbleweed Sep 04 '13 at 04:55
  • Yes @JonasMeyer, I missed the definition. Thank you. So now I got the proof correctly? – Tumbleweed Sep 04 '13 at 04:56
  • There is a small missing step, which is that the group does have a set of generators. It is cleaner to write as in the first sentence of my first comment. – André Nicolas Sep 04 '13 at 05:13

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Your proof is correct: If there are infinitely many distinct generators, there are infinitely many distinct group elements.

For an alternative (simpler?) proof, just note that $H = \langle H \rangle$.