Questions tagged [solvable-groups]
A solvable group is a group whose derived series terminates in the trivial subgroup.
56 questions
23
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2
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Solvable groups that are linear over $\mathbb{C}$ but not over $\mathbb{Q}$?
Let $\Gamma$ be a finitely generated finitely presented virtually solvable group. Assume that there exists an injective representation $\Gamma \to \operatorname{GL}_n(\mathbb{C})$. Is it true that ...
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votes
2
answers
1k
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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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18
votes
1
answer
752
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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 ...
15
votes
1
answer
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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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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 ...
- 153
13
votes
1
answer
370
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Factorizing groups into a product of solvable subgroups
Does every finite group $G$ have a factorization $G=H_1\cdots H_k$ where the $H_i$ for $1\le i\le k$ are solvable subgroups of $G$ and $|G|=|H_1|\cdots |H_k|$ (equivalently, every element of $G$ is ...
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12
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5
answers
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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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11
votes
1
answer
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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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11
votes
1
answer
499
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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 ...
- 1,173
10
votes
4
answers
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Conjugation by elements of subgroups
Let $G$ be a group generated by a conjugacy class $C$. I am interested in studying this property:
for every $x,y\in C$ there exists $h\in \langle x,y\rangle$ such that $y=hxh^{-1}$.
Basically the ...
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9
votes
1
answer
897
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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 ...
- 1,173
9
votes
1
answer
321
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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 ...
- 2,519
9
votes
0
answers
445
views
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 ...
- 44.4k
8
votes
2
answers
479
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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 $n$ ...
8
votes
0
answers
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Groups having exactly two non real-valued irreducible characters
This is an enlarged version of my question on MSE. It was suggested I ask here instead.
Suppose the finite group $G$ has exactly two conjugacy classes that are not self-inverse (a conjugacy class is ...
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votes
2
answers
1k
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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 ...
- 11.1k
7
votes
1
answer
256
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Krasner–Kaloujnine universal embedding theorem for finitely generated groups?
The Krasner–Kaloujnine universal embedding theorem states that any group extension of a group $H$ by a group $A$ is isomorphic to a subgroup of the regular wreath product $A \operatorname{Wr} H$. When ...
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votes
4
answers
1k
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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 ...
- 151k
5
votes
2
answers
377
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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 ...
- 1,702
5
votes
1
answer
221
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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\...
- 11.3k
5
votes
3
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579
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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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5
votes
0
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508
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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\...
- 12.6k
4
votes
4
answers
318
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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=...
- 1,625
4
votes
2
answers
500
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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 ...
- 537
4
votes
2
answers
559
views
Groups whose derived length is logarithmic in the order?
Is there a class of solvable groups $G$ having a derived length $O(\log\lvert G\rvert)$?
See Wikipedia for the definition of Big-Oh ($O$) and the definition of derived series of a group.
Any help ...
- 217
4
votes
1
answer
417
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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 ...
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4
votes
1
answer
455
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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 ...
- 11.3k
4
votes
2
answers
1k
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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 ...
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4
votes
1
answer
165
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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 ...
- 4,432
4
votes
0
answers
209
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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 ...
- 1,433
4
votes
0
answers
260
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\...
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4
votes
0
answers
200
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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 ...
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votes
0
answers
347
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 ...
- 1,324
4
votes
0
answers
124
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 ...
- 4,432
4
votes
0
answers
177
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 ...
user82740
3
votes
4
answers
757
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 ...
3
votes
3
answers
352
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 ...
- 215
3
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1
answer
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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 ...
- 415
3
votes
1
answer
260
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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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3
votes
1
answer
504
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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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3
votes
1
answer
215
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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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3
votes
1
answer
156
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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 ...
- 415
3
votes
1
answer
176
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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 ...
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2
votes
2
answers
304
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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 ...
- 11.3k
2
votes
1
answer
132
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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}(...
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2
votes
1
answer
265
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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 ...
- 1,805
2
votes
1
answer
401
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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 ...
- 2,143
2
votes
1
answer
121
views
Semi-direct decomposition of a solvable Lie group
(This is a cross-post from this MSE question)
I am searching for a reference or proof to the following fact (asserted at the top of page 2 here).
Let $G$ be a connected, solvable Lie group. Then $G = ...
- 185
2
votes
0
answers
79
views
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 ...
2
votes
0
answers
60
views
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 ...
- 21