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**tonio** Take for instance $\displaystyle S=\{s_1,s_2,s_3\}\subset G$ , for some group $\displaystyle G$. Then every elements of $\displaystyle G$ is the product of a finite string of elements of $\displaystyle S$ AND their inverses.

Among these elements we have $\displaystyle \{1\mbox{ (the empty product of elements of S) }\,,\,s_2\,\ldots\,,\,s_1s_2\,,s_1s_2s_3\,\ldots\}$

The above products are all in G, and they're in 1-1 correspondence with the subsets of $\displaystyle S:\,\,\{\emptyset\,,\,\{s_2\}\,\ldots\,\{s_1,s_2\} \,,\,\{s_1,s_2,s_3\}\,,\ldots\}$.

Note that as elements of $\displaystyle G$, ALL the above strings of elements of $\displaystyle S$ are different, otherwise the generating set wouldn't be minimal.

Tonio