Consider the additive group of integers as an example as mentioned at the bottom of the Wikipedia article. There are two generating sets that are mentioned; The set consisting of the number 1, {1}, and the two-element set, {3,5}. So when we talk about the number of generators of a group do we mean the number of different elements within every generating set, all added together. So above we've counted 3 generators so far, 1,3, and 5 – although there is clearly more than this in total since we haven't mentioned all the generating sets of the group.
As another example, in SU(N) we have $N^{2}-1$ generators. Would this be the total number of generating matrices that we'd get if we counted every single generating set?
On a similar note, would we only count a generator once if it appeared in more than one generating set? So for example if some group had generating sets {1,5}, {5,9}, {3,4,12}, would we say this group had 6 generators, rather than 7?
Best Answer
A group $G$ typically have many different generating sets, that is subsets $S\subseteq G$ with $\langle S\rangle = G$, and among these are sets of different cardinality (for example, trivially $S=G$ is a generating set). Therefore it makes little sense to speak of the number of generators of a group (or even of the set of generators).
We do speak of the free group $F_S$ generated by the set $S$ and for such a free group, the set $S$ is a canonical choice of generators. Nevertheless, the free group over $S=\{a,b\}$ is also generated by $\{a,b,1,a^{-2},bab\}$. In general, when we speak of a group generated by $n$ elements, we mean a group $G$ that allows an epimorphism $f\colon F_S\to G$ where $|S|=n$, typically given by a presentation $G=\langle a_1,\ldots,a_n\mid\ldots\rangle$. Depending on the author, it may additionally be understood that $f|_S$ is injective, that is that the generators of $G$ obtained this may are actually an $n$-element set. But I guess it is often not the case that this distinction is made.
In summary, "a group generated by $n$ elements" should usually be more precisely called "a group that has at least one generating set of at most $n$ elements". For example, a cyclic group is a group generated by a single element. We do however also count the trivial group as cyclic, even though it can in fact be generated by zero elements.