Questions tagged [solvable-groups]
A solvable group is a group whose derived series terminates in the trivial subgroup.
49
questions
2
votes
0
answers
66
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 ...
4
votes
0
answers
161
views
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 ...
- 817
15
votes
1
answer
789
views
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
18
votes
1
answer
690
views
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 ...
-1
votes
2
answers
279
views
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 ...
- 151
2
votes
1
answer
132
views
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}(...
- 123
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 ...
- 33
4
votes
1
answer
389
views
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 ...
- 877
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\...
- 985
3
votes
0
answers
144
views
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 ...
- 8,461
2
votes
0
answers
54
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
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 ...
- 1,657
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 $\Bbb C$.
- 3,001
3
votes
1
answer
121
views
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 ...
- 4,334
9
votes
1
answer
278
views
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 ...
- 2,459
1
vote
1
answer
87
views
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....
- 243
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 $n$ ...
- 25.1k
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 ...
- 178
9
votes
0
answers
407
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 ...
- 41k
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 ...
- 545
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 ...
- 1,274
4
votes
0
answers
120
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 ...
- 4,334
2
votes
1
answer
373
views
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
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 ...
- 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, ...
- 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 ...
- 395
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 ...
- 395
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 ...
- 395
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 ...
- 51
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 ...
- 1,662
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 ...
- 1,153
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 ...
user82740
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 \...
- 11.2k
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\...
- 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 ...
- 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 ...
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 ...
- 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 ...
- 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,
$$R_n = \big(x_1+x_2\...
- 11.5k
5
votes
4
answers
1k
views
The Icosahedron Equation
$$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 ...
- 148k
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=...
- 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 ...
- 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 ...
- 1,153
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 ...
- 215
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!)
...
- 148k
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 ...
- 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 ...
- 11k
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 ...
- 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 ...
- 10.4k