Questions tagged [solvable-groups]
A solvable group is a group whose derived series terminates in the trivial subgroup.
66 questions
1 vote
0 answers
76 views
Generating a finite non-solvable group with an element and its conjugate [migrated]
It is known that for any finite simple group $G$ there exist two elements $a,b\in G$ such that $a$ and $b$ are conjugates in $G$ and $\langle a, b \rangle=G$. My question: Is it true for any finite ...
- 381
12 votes
1 answer
281 views
A query regarding maximal subgroups of a finite non-solvable group
This is some kind of continuation of an earlier MO post: Existence of maximal subgroups of even order which are not normal It appeared in a comment in the above post. I believe the following is true: ...
- 381
7 votes
2 answers
403 views
A generalization of Hall theorem
Question. Let $G$ be a finite group of order $n$, and let $d$ be a divisor of $n$ such that $|\pi(d)|\geq 3$. Suppose that for every proper subset $A \subseteq \pi(d)$ (where $\pi(d)$ denotes the set ...
- 121
5 votes
1 answer
347 views
Why do we care about residually solvable/nilpotent groups?
I'm interested in a certain property $X$. In the introduction to basically every paper on $X$ there's a paragraph that goes something like: $X$ is related to residually-solvable groups in this way, ...
- 491
1 vote
0 answers
69 views
Open solvable subgroups of non-Archimedean Lie groups
Let $K$ be a local non-Archimedean field and $G$ a closed nondiscrete subgroup of $\mathrm{GL}_n(K)$, or a $K$-Lie analytic group (though I am primarily interested in linear groups). Is the following ...
- 11
9 votes
1 answer
308 views
Are metabelian groups quasi-isometrically 2-generated
The number of generating elements can vary fairly wildly when looking at finite index subgroups. In a discussion (inspired by various results about embedding group of some sort in a 2-generated group ...
- 4,706
11 votes
2 answers
448 views
Actions of finitely generated solvable groups on sets where every element has all finite orbits
Let $S$ be a finitely generated solvable group acting transitively on a set $X$ such that for each $s \in S$, every orbit $\langle s \rangle x$ (for $x \in X$) is finite. Does it follow that $X$ is ...
- 113
1 vote
0 answers
115 views
Solvable groups such that all but one $p$-core are contained in the center $Z(G)$
Let $G$ be a finite solvable group with $O_2(G)=1$, $O^{2'}(G)=G$ ( i.e. $G$ is the smallest normal subgroup with odd index) and suppose that there exist one and only one (odd) prime $p$ such that $...
2 votes
0 answers
77 views
Why is a minimal normal subloop of a solvable finite Moufang loop Abelian?
I am looking for a proof of the following "well-known" Fact: Every minimal normal subloop of a solvable finite Moufang loop is an Abelian group. For groups this fact is indeed well-known ...
- 44.3k
3 votes
1 answer
198 views
Example of a finitely generated metabelian group whose Fitting subgroup is not nilpotent
It is known that the Fitting subgroup of a finitely generated polycyclic-by-finite group is nilpotent, but this statement is not true for the solvable group. It is clear that both Lamplighter groups ...
- 1,063
8 votes
0 answers
241 views
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 ...
- 967
2 votes
1 answer
195 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 = ...
- 235
7 votes
1 answer
397 views
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 ...
- 860
13 votes
1 answer
438 views
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 ...
- 967
5 votes
2 answers
720 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 ...
- 227
24 votes
2 answers
1k views
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 ...
- 1,420
10 votes
4 answers
1k views
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 ...
- 477
2 votes
0 answers
105 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
261 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 ...
- 2,171
15 votes
1 answer
1k 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
800 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 ...
0 votes
2 answers
429 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 ...
- 227
2 votes
1 answer
169 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
238 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
478 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 ...
- 891
4 votes
0 answers
288 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\...
- 1,011
4 votes
0 answers
266 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 ...
- 11.5k
2 votes
0 answers
86 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
2 votes
1 answer
331 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 ...
- 1,923
1 vote
0 answers
224 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 $\Bbb C$.
- 3,119
4 votes
1 answer
233 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 ...
- 4,706
9 votes
1 answer
373 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 ...
- 2,549
1 vote
1 answer
154 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
573 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 $n$ ...
- 27.6k
11 votes
1 answer
300 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
515 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 ...
- 46k
15 votes
1 answer
676 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 whether ...
- 545
4 votes
0 answers
423 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
161 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 ...
- 4,706
2 votes
1 answer
437 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
256 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
717 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
220 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 ...
- 415
1 vote
0 answers
91 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 ...
- 415
3 votes
1 answer
1k 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 ...
- 415
1 vote
1 answer
269 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
469 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 ...
- 1,782
11 votes
1 answer
549 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 order ...
- 1,193
4 votes
0 answers
199 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
22 votes
2 answers
1k 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 \...
- 11.4k