For $n=1$ the answer is "yes." -- A group is abelian iff its generators commute. Let $G_0=G$ be a group and let it be generated by $X_0=X$. For each $n>0$ let $G_n=[G_{n-1},G_{n- …

Suppose $G$ is a connected Lie group whose radical is $R$. It is known that the solvable group $R$ can always be decomposed as $R=UT$ where $U$ is a simply-connected normal subgrou …

This is an extension of this MSE question, in which I asked whether there was a counterexample to the following statement, Conjecture. If a finite group $G$ contains a $\lbrace …

I have two questions about the class of integer-coefficient polynomials all of whose roots are rational. I asked this at MSE, but it attracted little interest (perhaps because it i …

Suppose that $\pi:Y \to Y'$ is a regular covering of closed, connected, orientable three-manifolds and let $G$ be the deck transformation group. Furthermore, suppose that $Y$ is a …

This question concerns finite groups. It is a well-known fact that every subgroup of a solvable group must again be solvable; this is easily proven by looking at the derived serie …

Consider a semi-direct product $\mathbb{Z}^2\rtimes_A\mathbb{Z}$, where $A\in SL_2(\mathbb{Z})$ and $|Tr(A)|>2$. It is clear that it is isomorphic to a lattice in the 3-dimensional …

For a given group presentation of a group(finitely presented), I want to check whether it is residually solvable or not. Is there any good way to do it? Actually, I'm curious whet …

The basic question is whether there is a notion of chief factor of a connected solvable algebraic group that matches my intuition. A few smaller assertions are sprinkled through t …