Questions tagged [solvable-groups]
A solvable group is a group whose derived series terminates in the trivial subgroup.
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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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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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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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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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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 ...
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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 ...
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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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2
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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 ...
- 1,234
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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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answer
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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 ...
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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 ...
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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 ...
user82740
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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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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\...
- 11k
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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 ...
- 11k
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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 ...
- 11k
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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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0
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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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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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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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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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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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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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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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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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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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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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