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Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either: 1) Let $\mathcal{L}$ ...

Is there a constant $K \in \mathbb{N}$ such that for every finite solvable group $G$, there exists a nilpotent subgroup $N \leq G$, and a subset $S \subseteq G$ with $|S| \leq K$, and $\langle N,S\...

The Lazard Correspondence is often phrased (for simplicity) for $p$-groups of nilpotence class $c < p$, but it works more generally whenever every 3-generated subgroup has nilpotence class $< p$,...

Let $p$ be a prime number. Is there a finite nonabelian $p$-group $G$ such that any finite epimorphic $2$-generated image of $G$ is abelian?

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z_{p^{2}}}\rtimes \mathbb{Z_{p^{}}}) ...

Let $F$ be a nonabelian finitely generated free pro-$p$ group, $H \leq_c F$ of infinite index. Denote by $\{F_n\}_{n \in \mathbb{N}}$ the lower central series of $F$, and set $r_n = [F : F_nH]$. Is ...