5 problems

Let $G$ be a finitely generated perfect group, meaning $G=[G,G]$, the commutator subgroup of $G$. Wiegold's normal generation conjecture. $G$ can be normally generated by a single…

Let $G$ be a torsionfree discrete group, and let $g_1,\dots,g_k$ be elements whose normal closure is $G$. The first $\ell^2$-Betti number of $G$ is denoted by $\beta_1^{(2)}(G)$. N…

Let $G_1,\ldots,G_n$ be finite groups, and let $G_1 * \cdots * G_n$ denote their free product. A group is normally generated by $k$ elements if its normal closure can be generated…

Let $G$ be a torsion free discrete group, or a $p$-group for a prime $p$, and let $g_1,\dots,g_n$ be arbitrary elements of $G$. Denote by…

Lennox–Wiegold conjecture. Every finitely generated perfect group has killing number $1$; equivalently, every finitely generated perfect group is the normal closure of a single ele…