Questions tagged [solvable-groups]

A solvable group is a group whose derived series terminates in the trivial subgroup.

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Is the continued fraction of a constructible number special in some way?

Rationals have finite CF and quadratic have periodic CF. CF in turn can be represented in terms of the modular group SL2(Z), e.g. using the standard generators S(z)=-1/z and T(z)=z+1. On the other ...
Lucian Ionescu's user avatar
4 votes
0 answers
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A different approach to proving a property of finite solvable groups

Edit: I'd be happy to hear any vague thoughts you might have, however far they may be from a complete solution! I asked this on math.stackexchange a couple of days ago, but it didn't attract any ...
semisimpleton's user avatar
15 votes
1 answer
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Is the infinite product of solvable groups amenable?

I am interested in the amenability properties of infinite products of solvable groups. The following facts are well-known: Any solvable group is amenable. The class of solvable groups is closed under ...
Asgar's user avatar
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18 votes
1 answer
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Is solvability semi-decidable?

Let $G = \langle A \mid R \rangle$ be a finitely presented group, given by a finite presentation. If $G$ is abelian, then we can verify this fact: simply verify the fact that $[a, b] = 1$ for all ...
Carl-Fredrik Nyberg Brodda's user avatar
-1 votes
2 answers
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Splitting of a finite group with no abelian subfactor in composition series

Let $G$ be a finite group with no abelian subfactor in its composition series. Is $G$ obtained from simple groups by iterating semidirect products? (Initially it was asked whether $G$ is a direct ...
Jins's user avatar
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2 votes
1 answer
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Element that is in $\phi^{-1}(Z(F (G/F(G)))$

I'm studying an article but I'm not able to understand one of his statements. I have the following hypotheses: $G$ is a solvable group with trivial center, $J=\phi^{-1}(F(G/F(G)))$ and $J_2=\phi^{-1}(...
Thiago Luiz's user avatar
3 votes
1 answer
152 views

Example of a supersolvable Lie group/algebra whose nilradical does not have a complement

What is an example of a real solvable simply-connected Lie group $G$ whose nilradical does not have a complement (that is, $G$ is not a semidirect product of the nilradical and another subgroup)? Is ...
Guest7819's user avatar
4 votes
1 answer
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Is a solvable group satisfying a semigroup law?

Let $S$ be the free semigroup on the set $\{x_1,\ldots ,x_n\}$, where $n$ is a positive integer. Suppose that $\mu=\mu (x_1,\ldots ,x_n)$ and $\nu = \nu (x_1,\ldots ,x_n)$ are two elements in $S$. We ...
mahdi meisami's user avatar
4 votes
0 answers
256 views

A big class of finite groups

During my researches, I've obtained a class of finite groups as follows. Let $\mathcal{C}$ be the class of all finite groups $G$ such that for every factorization $|G|=ab$ there exists a subgroup $H\...
M.H.Hooshmand's user avatar
3 votes
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Derived length in linear groups

If $G$ is a group let $(G^{(m)})_{m \geq 0}$ be the derived series. If there is some $m$ such that $G^{(m+1)} = G^{(m)}$, call the smallest such $m$ the derived length of $G$. I am interested in ...
Sean Eberhard's user avatar
2 votes
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Is there always a purely real representative for a metrized solvable Lie group?

Alekseevski proves for Heintze groups (a special class of solvable Lie groups) that any such group admits a (left-invariant) metric which is isometric to a purely real Heintze group (again equipped ...
Burns Healy's user avatar
3 votes
1 answer
240 views

Does the sequence (Number of groups of even order $\le n$) / (Number of groups of order $\leq n$) converge? If not, what are its cluster points?

I recently gave an undergraduate course on group theory (which is not entirely my field of expertise, so the following questions might have a well-known answer of which I am simply unaware). As I was ...
The Thin Whistler's user avatar
1 vote
0 answers
131 views

Is every connected solvable group Borel?

Is every connected solvable algebraic group a Borel subgroup of a reductive group? If a counterexample exists, I would ideally like it to be over $\Bbb C$.
Avi Steiner's user avatar
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Centre of solvable locally nilpotent groups

This question is motivated by two examples of locally nilpotent groups which I came across (see below). Question: Given an infinite solvable and locally nilpotent group $G$, does $G$ have an infinite ...
ARG's user avatar
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9 votes
1 answer
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Subgroups of infinite solvable groups

I'm looking for results of the form "every infinite solvable group contains <...> as a subgroup". Specifically, I believe: If $G$ is infinite solvable, finitely generated and not ...
grok's user avatar
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1 vote
1 answer
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Infinite pro-$p$ group of finite solvable length and finite coclass

I was reading about infinite pro-$p$ groups of finite coclass from the book "The Structure of Groups of Prime Power Order" by Leedham-Green and McKay. I asked this question in math....
usermath's user avatar
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8 votes
2 answers
439 views

Abundancy index and non-solvable finite groups

Let $\sigma$ be the sum-of-divisors function. A number $n$ is called abundant if $\sigma(n)>2n$. Note that the natural density of the abundant numbers is about $25 \%$. The abundancy index of $n$ ...
Sebastien Palcoux's user avatar
11 votes
1 answer
223 views

Does $\chi(1)^2=|G:Z(G)|$ for irreducible character of a finite group $G$ imply $G$ is solvable?

In "Character Theory of Finite Groups" I.M. Isaacs mention the following conjecture: It is only possible in a solvable group $G$ to have $\chi(1)^2=|G:Z(G)|$ with $\chi \in$ Irr$(G)$. Is this ...
Anton B's user avatar
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Which finite solvable groups have solvable automorphism groups?

Is it possible to give a reasonable description of those finite solvable groups $G$ such that $A = {\rm Aut}(G)$ is also solvable? The central case to deal with is that in which $G$ is a $p$-group of ...
Geoff Robinson's user avatar
15 votes
1 answer
605 views

Does $\mathbb{Q}$ embed into a finitely generated solvable group?

Does $\mathbb{Q}$ embed into a finitely generated solvable group? I've checked that $\mathbb{Q}$ is not a subgroup of any finitely generated metabelian group. I don't know how to show this (or ...
Josh F's user avatar
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4 votes
0 answers
283 views

Any way around Abel's impossibility theorem?

Abel's impossibility theorem states that the roots of a general polynomial (of degree 5 or higher) cannot be written using arithmetic operations and radicals. Radicals are solutions of a specific ...
MCH's user avatar
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4 votes
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Abelian-by-cyclic subgroups of exponential growth solvable groups

I am currently looking for a reference to a proof (or counterexample) to the following statement: Statement: Assume $G$ is a finitely generated solvable group of exponential growth, then there is a ...
ARG's user avatar
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2 votes
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Questions about a finite solvable group

These questions are by Moshe Newman Let $G$ be a finite solvable group of derived length $d$, with the property that every proper subgroup and every proper quotient of $G$ has derived length less ...
David S. Newman's user avatar
3 votes
1 answer
199 views

Commutator length in connected solvable Lie groups

Let $G$ be a connected solvable Lie group and let $H$ denote ist commutator subgroup. By definition, every element $g \in H$ can be written as a product of commutators and the minimal number of ...
Lyonel's user avatar
  • 97
5 votes
3 answers
487 views

Solvable Lie algebra application

I am starting to study Lie algebras and when I reached the notion of solvable Lie algebra, I tryed to find concrete applications ( in physics for exemple) and I couldn't find one. For exemple, ...
poloC's user avatar
  • 153
3 votes
1 answer
129 views

Bounding the derived length of a solvable group given the degrees of the irreducible monomial characters

Much is known about the derived length of a solvable group given the degrees and cardinality of the set of degrees of the irreducible characters. Martin Isaacs and Donald Passman pretty much started ...
Joakim Færgeman's user avatar
1 vote
0 answers
54 views

Irreducible characters of a semi-direct product with a p-group

Suppose G is a semi-direct product of P with H where P is a (non-abelian) p-group and G is solvable. I wonder what can be said about the irreducible characters of G given information about the ...
Joakim Færgeman's user avatar
3 votes
1 answer
771 views

Conditions for a solvable group to have a non-trivial center

I am working on a problem in character theory where I try to bound the derived length of a solvable group using information about its characters. In my specific case, it will be extremely helpful for ...
Joakim Færgeman's user avatar
1 vote
1 answer
221 views

Portability of Thompson theorem about solvability to Moufang loops

Say we have a finite Moufang Loop $Q$, $|Q|<\infty$. There is a theorem proved by Thompson that states: Group $G$, $|G|<\infty$ is solvable $\iff$ $\forall a, b \in G \langle a, b\rangle$ is ...
Evgeny's user avatar
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5 votes
2 answers
366 views

Does the group G(K) have a cocompact solvable closed subgroup?

Let $K$ be a (locally compact) local field and $G$ be a linear algebraic $K$-group. Does the topological group $G(K)$ have a cocompact solvable closed subgroup? If $\mathrm{char}(K)=0$, it is true ...
m07kl's user avatar
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11 votes
1 answer
459 views

Are all sneaky groups products of Frobenius and 2-Frobenius groups?

I've been stuck thinking about this for a while. Def. Let $G$ be a finite solvable group whose order is divisible by only three primes: $p,q,$ and $r$. Suppose that $G$ has cyclic subgroups of ...
Alexander Gruber's user avatar
4 votes
0 answers
156 views

Is there any probabilistic characterization for generalized solvable groups?

References: This question is inspired by a conjecture of Alon Amit that is solved by Miklós Abért, Nikolay Nikolov and Dan Segal in the following papers: (1) On the probability of satisfying a word in ...
user avatar
21 votes
2 answers
968 views

Is there a big solvable subgroup in every finite group?

Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H \...
Pablo's user avatar
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5 votes
1 answer
218 views

Finite solvable groups are generated by a nilpotent subgroup + K elements?

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\...
Pablo's user avatar
  • 11.2k
3 votes
1 answer
244 views

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 ...
Pablo's user avatar
  • 11.2k
3 votes
4 answers
708 views

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 ...
Bhaskar Vashishth's user avatar
4 votes
1 answer
424 views

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 ...
Pablo's user avatar
  • 11.2k
2 votes
2 answers
267 views

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 ...
Pablo's user avatar
  • 11.2k
5 votes
0 answers
483 views

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 = \big(x_1+x_2\...
Tito Piezas III's user avatar
5 votes
4 answers
1k views

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 ...
Joseph O'Rourke's user avatar
3 votes
4 answers
312 views

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 $X_n=...
Matt Brin's user avatar
  • 1,555
1 vote
2 answers
762 views

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 ...
Li Yu's user avatar
  • 11
9 votes
1 answer
853 views

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 ...
Alexander Gruber's user avatar
3 votes
3 answers
340 views

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 ...
Tye Lidman's user avatar
12 votes
5 answers
2k views

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!) ...
Joseph O'Rourke's user avatar
3 votes
1 answer
476 views

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 ...
ARupinski's user avatar
  • 5,161
6 votes
2 answers
1k views

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 ...
Alain Valette's user avatar
4 votes
2 answers
469 views

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 ...
hopflink's user avatar
  • 537
4 votes
2 answers
1k views

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 ...
Jack Schmidt's user avatar
  • 10.4k