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### Schreier's formula and supersolvable groups

A finitely generated profinite group $G$ is said to satisfy Schreier's formula if for every open subgroup $L \leq_o G$ we have $d(L) = (d(G)-1)[G:L] + 1$. Here $d$ stands for the smallest cardinality ...

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### Lucido's three prime lemma

Let G be a finite solvable group. If p,q,r are distinct primes dividing |G|, then G contains an element of order the product of two of these three primes.
This is lucido's three prime lemma. I ...

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### Generators of Sylow subgroups

Is there a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for each finite supersolvable group $G$, and a Sylow subgroup $S \leq G$ we have $d(S) \leq f(d(G))$?
Here $d(H)$ denotes the ...

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### Number of generators of the commutator

Can one find a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for every finite supersolvable group $G$ we have: $d(G') \leq f(d(G))$?
Here $d(K)$ is the cardinality of a minimal set of ...

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### On the peculiar Lagrange resolvent of the septic $7x^7+14x^4+7x^3-1=0$

Given an irreducible solvable equation $P(x)=0$ of prime degree $p>2$ with rational coefficients and $\zeta^p=1$, define the usual Lagrange resolvents of the roots $x_i$ as,
$$R_n = ...

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### The Icosahedron Equation

$$1728 V^5 + F^3 = E^2 \;.$$
Can anyone point me to a concise, modern derivation and explanation of
the significance of the icosahedron equation, more modern and
concise than Klein's description in ...

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### Can group solvability be detected from identities among the generators?

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 $G_n=[G_{n-1},G_{n-1}]$ and let ...

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### Decomposition of solvable Lie group

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 subgroup of $R$ and $T$ is ...

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### An extension of the converse to Hall's theorem.

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 $\lbrace p,q \rbrace$-Hall ...

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### When is a three-manifold deck transformation group solvable?

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 rational homology ...

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### Polynomials all of whose roots are rational

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 is not interesting!)
...

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### Generalization of a Result on Solvable Groups

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 series of a given ...

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### Lattices in SOL

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 solvable Lie ...

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### Is there any way to check whether a group is residually solvable?

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 whether the finitely ...

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### Are all connected solvable affine algebraic groups supersolvable?

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 the explanation, and ...