It is well-known that a finite group can be generated by arbitrary representatives of each conjugacy class, see e.g.: https://mathoverflow.net/questions/26979/generating-a-finite-group-from-elements-in-each-conjugacy-class
Now, for infinite groups this fails in general, for example the upper triangular matrices are a conjugate-dense subgroup of $GL_n(K)$ if $K$ is an algebraically closed field. So in this example there is some choice of representatives of conjugacy classes such that these elements do not generate the whole group. I am looking now for other examples:
- Is there an infinite group $G$ such that for any choice $S$ of representatives of conjugacy classes, the subgroup generated by $S$ is proper.
- As in 1. and additionally requiring that $G$ is finitely generated.
- As in 2. and additionally requiring that $G$ has finitely many conjugacy classes.