Questions tagged [solvable-groups]

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

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$G/F(G)$ is isomorphic to $X_1\times\cdots\times X_t$

I asked this question on math.stackexchange many hours ago, but haven’t got an answer. It was mentioned in a comment that the answer to my question is trivial, but I couldn’t see why. $G$ is a finite ...
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Fitting subgroup of a solvable group is a direct product of some elementary abelian $p$-groups

I saw a remark saying If $G$ is solvable, then ${\rm Out}(F(G))$ is isomorphic to a direct product of ${\rm GL}(n_i,p_i)$ (except for certain value of $n_i$ and $p_i$). I know the key is to prove ...
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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 ...
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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 ...
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1answer
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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 ...
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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 ...
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73 views

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 ...
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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 ...
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1answer
135 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 ...
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3answers
339 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, ...
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1answer
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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 ...
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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 ...
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1answer
295 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 ...
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1answer
194 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 ...
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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 ...
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1answer
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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 ...
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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 ...
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Gauss-Wantzel theorem, Fermat primes and solvability of S_n [closed]

Gauss-Wantzel theorem asserts that a polygon with $n$ sides is constructible if and only if $n$ is a product of a power of $2$ and distinct prime Fermat numbers, where the Fermat number of index $k$ ...
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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 \...
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1answer
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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\...
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1answer
182 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 ...
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4answers
617 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 ...
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1answer
297 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 ...
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2answers
243 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 ...
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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 = \big(x_1+x_2\...
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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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4answers
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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 $X_n=...
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2answers
689 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 ...
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1answer
749 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 ...
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3answers
318 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 ...
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5answers
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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!) ...
3
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1answer
438 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 ...
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2answers
843 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 ...
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2answers
415 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 ...
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2answers
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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 ...