16.87 (2006)

Let $\mathfrak{M}$ be a variety of groups and let $G_r$ be a free $r$-generator group in $\mathfrak{M}$. A subset $S \subseteq G_r$ is called a test set if every endomorphism of $G_r$ identical on $S$ is an automorphism. The minimum of the cardinalities of test sets is called the test rank of $G_r$. Suppose that the test rank of $G_r$ is $r$ for every $r \geqslant 1$.
$\qquad$ a) Is it true that $\mathfrak{M}$ is an abelian variety?
$\qquad$ b) Suppose that $\mathfrak{M}$ is not a periodic variety. Is it true that $\mathfrak{M}$ is the variety of all abelian groups?