7.57 (1980)

A set of generators of a finitely presented group $G$ that consists of the least possible number $d(G)$ of generators is called a basis for $G$. Let $r_M(G)$ be the least number of relations necessary to define $G$ in the basis $M$, and $r(G)$ the minimum of $r_M(G)$ over all bases $M$ for $G$.
$\qquad$ a) It is known that $r_M(G) \leqslant d(G) + r(G)$ for any basis $M$. Does there exist a finitely presented group $G$ for which the inequality becomes equality for some basis $M$?

Let $G_1$, $G_2$ be any non-trivial groups.
$\qquad$ b) Is it true that $r_{M_1 \cup M_2}(G_1 \ast G_2) = r_{M_1}G_1 + r_{M_2}(G_2)$ for any bases $M_1, M_2$ of $G_1, G_2$, respectively?
$\qquad$ c) Is it true that $r(G_1 \ast G_2) = r(G_1) + r(G_2)$?