I've been looking into finding the presentation of subgroups $H$ of $G$ from a known presentation of the group $G$.
I know that usually, removing generators but keeping the set of relations the same can give presentations of subgroups. However, I've heard that there are instances where a subgroup might actually require more generators than the group itself. Does anyone know of any concrete examples of this that they can explain/know of where I can find an explanation?
2026-03-25 01:15:05.1774401305
Presentations of subgroups
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This can happen even for finite groups: every finite group is a subgroup of a 2-generator group, namely a symmetric group. The simplest example of finite groups which requires more than two generators is $C_2 \oplus C_2 \oplus C_2$, where $C_2$ is cyclic of order 2. For infinite groups, it is a theorem Higman, Neumann and Neumann that every countable group is a subgroup of a group generated by two elements.
To find presentations of subgroups of a finitely presented group in practice, you may want to read about the Todd-Coxeter algorithm.