# Questions tagged [solvable-groups]

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

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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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### 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 ...
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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 vote
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### 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 $\Bbb C$.
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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 ...
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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 ...
1 vote
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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....
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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 $n$ ...
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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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### 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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### 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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### 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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### 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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### 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 ...
1 vote
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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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### 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 ...
1 vote
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### 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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### 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 word in ... 884 views

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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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### 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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### 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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### 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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