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?
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Sign up to join this communityI 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?
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 for a discussion. This is of course much stronger than generated by finite order.
The cardinality of the set of groups generated by three elements of order two is equal to the cardinality of the set of countable groups, i.e., the cardinality of the continuum.
Every countable group embeds in a 2-generator group by the Higman-Neumann-Neumann embedding theorem, and the two generators can be chosen to have infinite order.
Since a 2-generator free group embeds into the free product of three cyclic groups of order 2, it is easy to show (starting from the HNN-embedding theorem) that every countable group embeds in a group generated by three elements of order two.