Changes

A subset $S$ of a group $G$ is termed a '''generating set''' if it satisfies the following equivalent conditions: * For any element $g \in G$, we can write: $g = a_1a_2 \ldots a_n$ where for each $a_i$, either $a_i \in S$ or $a_i^{{fillin}-1}\in S$ (here, the $a_i$s are not necessarily distinct).* If $H$ is a [[proper subgroup]] of $G$ (i.e. $H$ is a [[subgroup]] of $G$ that is not equal to the whole of $G$), then $H$ cannot contain $S$.* Consider the natural map from the free group on as many generators as elements of $S$, to the group $G$, which maps the freely generating set to the elements of $S$. This gives a surjective homomorphism from the free group, to $G$.