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17.98 (2010)

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A variety of groups is said to be small if it contains only countably many non-isomorphic finitely generated groups.
$\hspace{0.8cm}$ a) Is it true that if $G$ is a finitely generated group and the variety $\text{Var}(G)$ it generates is small then $G$ satisfies the maximal condition on normal subgroups?
$\hspace{0.8cm}$ b) Is it true that a variety is small if and only if all its finitely generated groups have the Hopf property?

Contributor: P. M. Neumann

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