I just start learning about presentations of groups. I would like to ask a question. If G is a finitely-presented group with a presentation [S,R] (S denotes the set of generators, R denotes the set of relators) such that the cardinality of S is minimal in the class of all presentations of G. Is it true that the cardinality of S equals the minimal cardinality of generating set of G? If not true in general, is it true for finite groups?
If $S$ is a generating set, then $\langle S,R\rangle$ gives a presentation if we let $R$ be the set of all relations between elements of $S$.
In other words, the collection of sets $S$ that show up in presentations is identical to the collection of sets $S$ that show up as generating sets—so certainly the minimum cardinality is equal.