# How many finitely-generated-by-elements-of-finite-order-groups are there?

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?

• Most likely, continuum, for the same reason that there is continuum of finitely generated groups. – Moishe Kohan Apr 15 at 23:56
• My guess is that every f.g. group embeds into a group generated by finitely many torsion elements. Since each such group contains countably many f.g. subgroups but there are uncountably many f.g. groups, this would imply that there are uncountably many groups generated by finitely many torsion elements. – Andy Putman Apr 16 at 0:01
• Is it possible to compute the cardinality of the set of groups generated by $s,t$ with $s^2=t^3=1$? – Gerry Myerson Apr 16 at 0:12
• Grigorchuk constructed uncountably many groups of intermediate Growth generated by 3 involutions that I believe form uncountably many isomorphism classes but I have to double check – Benjamin Steinberg Apr 16 at 0:55
• @AndyPutman yes: every virtually (free non-abelian) group is SQ-universal. In particular this applies to $\mathrm{PSL}_2(\mathbf{Z})=\langle s,t\mid s^2=t^3=1\rangle$. It's even easier to directly show the latter has continuum many normal subgroups (given that the isomorphism relation between quotients of a given finitely generated group has countable casses, it implies continuum many isomorphism classes). – YCor Apr 16 at 6:36

• The last paragraph needs some elaboration. Indeed if $G\subset F_2/N$ and $F_2\subset H$ with $H$ free product of three cyclic groups of order 2 ($F_2$ has index 2 in $H$), $N$ is possibly not normal in $H$. – YCor Apr 16 at 18:10
• point 37, p.66 of Topics in Geometric Group Theory By Pierre de la Harpe tinyurl.com/ahzwd8ph suggests we only need $\mathbb{Z}_2\ast \mathbb{Z}_3$. – JP McCarthy Apr 17 at 7:21
• Yes, either $\mathbb{Z}_2*\mathbb{Z}_2*\mathbb{Z}_2$ or $\mathbb{Z}_2*\mathbb{Z}_n$ for any $n>2$ will do. – IJL Apr 18 at 10:49