I do not know where this question is on the trivial to intractable spectrum.


Consider the set of isomorphism classes of groups finitely generated by elements of finite order. What is the cardinality of this set?



How many finitely-generated-by-elements-of-finite-order-groups are there?
Grigorchuk showed that there are uncountably many growth degrees of 2-generated infinite p-groups for any prime p. Hence there are uncountably many isomorphisms classes and even quasi-isometry classes of finitely generated torsion groups. See page 116 of [Grigorchuk - On the Gap Conjecture concerning group growth](https://doi.org/10.1007/s13373-012-0029-4) for a discussion. This is of course much stronger than generated by finite order.


