For questions on solvable groups, their properties, and structure.

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35
votes
2answers
608 views

Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider $$G=\operatorname{Gal}(\mathbb{Q}(r_1,\...
14
votes
1answer
357 views

When is $G \ast H$ solvable?

In a proof that the lamplighter group $\mathbb{Z}_2 \wr \mathbb{Z}$ is not finitely presented, I showed that $\mathbb{Z}_2 \ast \mathbb{Z}$ is not solvable. More precisely, one can prove that the ...
13
votes
4answers
715 views

If a finite group $G$ is solvable, is $[G,G]$ nilpotent?

If $G$ is a connected solvable Lie-group, then $[G,G]$ is nilpotent. The corresponding statement for Lie algebras follows from Lie's theorem, and it then follows from connected Lie groups by ...
13
votes
2answers
287 views

What can be said about two groups with isomorphic derived factors?

The third isomorphism theorem states that we can relate an isomorphic relation between two normal subgroups of a group $G$. My question is can we infer anything about the two groups structures itself ...
10
votes
2answers
231 views

Free groups in some classes?

I understand that the only free groups that are abelian are 1 and Z, hence a difference between 'free abelian groups' and 'abelian free groups'. Can someone please tell me what are the solvable free ...
9
votes
0answers
79 views

Cover and avoidance properties of chief factors: can we avoid exactly the ones we want?

If $G$ is a finite solvable group with chief series $1 = G_0 \leq \ldots \leq G_n = G$ (so each $G_i \unlhd G$ and if $G_{i-1} < H < G_i$ then $H$ is not normal in $G$) and $I \subseteq \{ 1,\...
8
votes
4answers
868 views

Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
8
votes
1answer
482 views

Finite groups with all maximal subgroups of prime power index are solvable?

It is well known that in finite solvable groups, any maximal subgroup has prime power index. Question: If $G$ is a finite group in which any maximal subgroup has prime power index, is $G$ ...
8
votes
1answer
878 views

Group of order $8p$ is solvable, for any prime $p$

Consider the following question: Show that a group $G$ of order $8p$ is solvable, for any prime $p$. I am kind of stuck, but here are my first attempts: I chose the series of subroups $G>H_8&...
8
votes
1answer
103 views

Solvable groups and orders of elements

I am trying to prove the following result. Let $G$ be a group containing elements $x$ and $y$ such that the orders of $x$, $y$, and $xy$ are pairwise relatively prime; prove that $G$ is not ...
7
votes
3answers
136 views

How can I tell if $x^5 - (x^4 + x^3 + x^2 + x^1 + 1)$ is/is not part of the solvable group of polynomials?

I have developed an interest in generalisations of the fibonacci sequence, from tribonacci sequence up to what I'll coin the 'infinibonacci' sequence. I'm aware that these nth-bonacci sequences ...
7
votes
1answer
272 views

Solvability by radicals of an equation of prime degree

For which prime $p$ the equation $x^p-p^px+p=0$ is solvable by radicals? I don't know how to solve this for primes $p\neq 2$, so any help is welcome.
7
votes
1answer
111 views

Is there any relation between Galois solvability and integrability of hamiltonian systems?

Galois theory provides a method and formalism to study solutions of polynomial equations and solvability. Dynamical Hamiltonian systems have a somewhat similar concept of integrability. Since many ...
6
votes
2answers
299 views

Is $(\mathbb{Z},+)$ a solvable group?

Ok, so abelian groups are solvable. And Thm II.8.5 of Hungerford says A group is solvable iff it has a solvable series. (The group may be finite or infinite.) However, I can't seem to find a ...
6
votes
3answers
964 views

Show that a group of order $p^2q^2$ is solvable

I am trying to prove that a group of order $p^2q^2$ where $p$ and $q$ are primes is solvable, without using Burnside's theorem. Here's what I have for the moment: If $p = q$, then $G$ is a $p$-group ...
6
votes
1answer
92 views

When does a formula for the roots of a polynomial exist?

My question is straightforward to pose: given a polynomial $f$ over a subfield of $\mathbb{C}$, are there conditions which guarantee the existence of a closed formula for the roots of $f$ in terms of ...
6
votes
1answer
108 views

If $G$ is solvable and $G/[G,G]$ is cyclic, can $G\times G$ be generated by 2 elements?

I doubt this is true, but I haven't found any small counterexamples (there are no counterexamples with $|G| < 1536, |G| \neq 768$): Suppose $G$ is finite, solvable, 2-generated, and $G/[G,G]$ ...
5
votes
4answers
468 views

Is a polynomial equation of degree $\ge 5$ not solvable by any way?

By Galois Theory an arbitrary polynomial equation of degree $\ge 5$ is not solvable using radicals, unlike the polynomial equation of second degree which is solvable by radicals (because of the ...
5
votes
2answers
408 views

Why is a group of order 135 nilpotent?

Why does order 135 imply nilpotent?
5
votes
1answer
1k views

Every minimal normal subgroup of a finite solvable group is elementary abelian

Show that every minimal normal subgroup of a finite solvable group is an elementary abelian $p$-group for some prime $p$. I'm stuck on this one, any idea is appreciated.
5
votes
1answer
239 views

Solvable by radicals or not?

Let $f=2x^5-5x^4+5\in \Bbb Q[x]$. Then, how to prove that it's not solvable by radicals? Since $f$ is solvable by radicals iff $Gal(f)$ is solvable and $Gal(f)\subset S_5$ (up to isomorphism), I try ...
5
votes
1answer
202 views

sylow basis of finite solvable groups

Let ‎‎$G‎‎$‎ be a‎ ‎finite ‎solvable non-$p$-group ‎and ‎‎$‎‎A$ ‎be a ‎‎‎‎maximal ‎subgroup ‎of ‎‎$G‎‎$‎. ‎ Therefore ‎$A‎‎$ ‎is ‎of ‎primary ‎index ‎‎$p^{n}‎ $‎‎, that is ‎$|G : A|=p^{n}‎$ ‎‎where ‎$‎...
4
votes
1answer
1k views

How to prove that a group of order $72=2^3\cdot 3^2$ is solvable?

Let $G$ be a group of order $$72=2^3\cdot 3^2$$ Without using Burnside's Theorem, how to show that $G$ is solvable? Atempt: If we can show that $G$ has at least one non-trivial normal subgroup $N$, ...
4
votes
1answer
519 views

Solvable and nilpotent groups, normal series and intuition

I'm reading Hungerford's algebra and I'm on Nilpotent and solvable groups chapter. Hungerford starts with: Consider the following conditions on a finite group G: i) G is the direct product ...
4
votes
2answers
409 views

Index of maximal subgroups of $p$-solvable groups

If $G$ is finite $p$-solvable group (every chief factor has order either a power $p$ or relatively prime to $p$) must every maximal subgroup have index either a power of $p$ or relatively prime to $p$?...
4
votes
2answers
113 views

Subgroups of smallest possible index in a solvable group

The following question appears in Isaacs' Finite Group Theory: 3B.15) (Berkovich) Let $G$ be solvable, and let $H<G$ be a proper subgroup having the smallest possible index in $G$. Show that $H\...
4
votes
1answer
107 views

Unique normal subgroups of every possible order

Can one characterize groups $G$ for which there is a unique normal subgroup of order $d$ for every divisor of the order of $G$? For example must they be solvable?
4
votes
1answer
96 views

A Galois Extension over a field of characteristic p

Hi! I've been having a bit of trouble with this question, namely the very last part. It's from a past paper for a Galois Theory course, in which I have an exam on Monday. Basically, I can show every ...
4
votes
1answer
111 views

Is there a simple construction of a finite solvable group with a given derived length?

Given some integer $n$, is there an easy way to construct a finite solvable group of derived length $n$? It would seem that given a solvable group of length $n-1$, one should be able to form the ...
4
votes
1answer
118 views

Groups of order $p^aq^br^c$ containing elements of order $pq$, $qr$, and $pr$, but not $pqr$

Let $G$ be a solvable group of order $p^aq^br^c$ (for distinct prime $p,q,r$) containing elements of orders $pq$, $qr$, and $pr$, but no element of order $pqr$. Furthermore, assume that $G$ is ...
4
votes
1answer
269 views

If $N$ and $G/N$ are virtually solvable, then $G$ is virtually solvable?

("Virtually solvable" means there's a solvable subgroup of finite index.) I know this statement holds if "virtually solvable" is replaced by "solvable", but I want to knock it down to the finite-...
4
votes
1answer
68 views

Are these groups solvable?

I am thinking of Baumslag-Solitar groups of type $BS(1,m)=\langle a,b \mid bab^{-1} = a^m\rangle$ as a prototype. We can think of them as follows: Start with an infinite cyclic group $\langle a\...
4
votes
1answer
69 views

Is “being solvable” a geometric property for linear algebraic groups?

Say $G$ is a solvable linear algebraic group over some field $k$ of characteristic 0. This means that its derived series eventually terminates with a 1. My question is: Is "being solvable" a ...
4
votes
1answer
155 views

Clarification on proof that all groups of order $< 60$ are solvable

I've manged to prove that all groups of order $< 60$ are solvable, using Burnside's theorem. However, I found an alternate proof here Question about solvable groups It states that: "Note that $A_5$...
4
votes
1answer
35 views

Group theory commutator and solvable groups

let G be a group such that it contains 2 members $a, b \in G$ that statisfy: $a = p^{-1} b p$ where $p \in G$ $a = q^{-1} [a,b]q $ where $q \in G$ $a,b,[a,b]\neq e$ where $[a,b]$ is the commutator ...
4
votes
0answers
122 views

Does there exist an infinite solvable group with no normal abelian subgroups?

This is impossible if $G$ is finite and solvable, because then $G$ has a (nontrivial) minimal normal subgroup $A$, which can be shown (using a trick) to be abelian. I'm trying to mimic the same ...
4
votes
1answer
72 views

solvable subalgebra

I want to show that a set $B\subset L$ is a maximal solvable subalgebra. With $L = \mathscr{o}(8,F)$, $F$ and algebraically closed field, and $\operatorname{char}(F)=0$ and $$B= \left\{\begin{pmatrix}...
3
votes
2answers
119 views

Why is a finite group, with a maximal subgroup which is abelian, soluble?

I've come across an exercise saying the following: If $G$ is a finite group which contains a maximal subgroup $M$ which is abelian, show that $G$ is solvable and that $G^{(3)}$ (the third term in the ...
3
votes
2answers
118 views

A proof that BS(1,2) is not polycyclic

I am looking for examples of finitely generated solvable groups that are not polycyclic. In Wikipedia Baumslag-Solitar group $BS(1,2)$ is an example. But how to prove this fact?
3
votes
3answers
73 views

Existence a certain subgroup of a group

‎Let ‎‎$G$ ‎be a‎ ‎finite group ‎such ‎that ‎‎$G=P\rtimes Q‎‎‎‎$ ‎where ‎‎$P\in {\rm Syl}_p(G)‎‎$;‎ ‎‎$‎‎P\cong \Bbb{Z}_p\times \Bbb{Z}_p‎$ ‎‎‎‎ and ‎$Q\in {\rm Syl}_q(G)$; ‎$‎‎|Q|=q$ (‎$‎‎p, q$ ‎are ...
3
votes
1answer
58 views

Finding a particular chief series for a supersolvable group

Let $G$ be a finite supersolvable group. Since it is supersolvable it has (by definition) some normal series $1=N_0 \le N_1 \le \cdots \le N_n=G$ such that each factor $N_i/N_{i+1}$ is cyclic. Also ...
3
votes
3answers
134 views

A well-known theorem of O. Schmidt

Prove that if all the proper subgroups of a finite group $G$ are nilpotent, then $G$ is soluble. How to I prove it? Thanks in advance.
3
votes
1answer
68 views

prove that $GL_2(\Bbb Z_3)$ is solvable

I need to prove that $GL_2(\Bbb Z_3)$ is solvable What I tried: I know that $GL_2(\Bbb Z_3)$ has $(3^2-1)(3^2-3) = 48 = 3 * 2^4$ elements. I know that $n_3 \in \{1,4,16\}$ and $n_2 \in \{1,3\}$ ...
3
votes
1answer
95 views

Characterization of solvable groups in terms of subgroups of certain orders?

In this question, the OP mentions the following result: a finite group $G$ is solvable if and only if $$\text{for all $n$ dividing $|G|$ such that $\gcd(\frac{|G|}{n},n)=1$, $G$ has a subroup order $...
3
votes
1answer
76 views

Clarification of Abel-Ruffini theorem statement

From what I've studied, Abel-Ruffini theorem states that we can't find all the roots of some polynomials with degree above 5, using only radicals and arithmetic operations. How does it imply that we ...
3
votes
1answer
124 views

Does the class of soluble groups whose $p$-length is $\leq 1$ for all $p$ form a saturated Fitting formation?

Let $\mathcal{F}$ be the class of all soluble groups $G$ such that the $p$-length $l_{p}(G) \leq 1$ for all primes $p$. How do I show ‎$\mathcal{F}$ is a saturated Fitting formation?
3
votes
1answer
32 views

show a group with prime order product is solvable

Is a group with order $16*17$ solvable? I know that from Burnside this is solvable since 2 and 17 are prime and 4 is greater than 0. However, I am not allowed to use it, so what should I do? ...
3
votes
1answer
126 views

Example of $A \le G$ solvable, $B \lhd G$ solvable, but $AB$ is not solvable

We'll denote $A \lhd G$ for $A$ a normal subgroup of $G$; and $A \leq G$ to mean $A$ is a subgroup of $G$. I don't know if it's a tricky question. But it seems strange to me. The question asks for an ...
3
votes
1answer
107 views

Index of maximal proper subgroup of a solvable group

This is problem 2.7.15 from Hungerford's Algebra: If $H$ is a maximal proper subgroup of a finite solvable group $G$, then $[G:H]$ is a prime power. If $G$ is abelian, then it's easy to show ...
3
votes
1answer
73 views

Is meta-$p$-nilpotent for all $p$ the same as meta-nilpotent?

I've been working on understanding solvable groups one prime at a time, and it has been very helpful. Perhaps this question is enjoyable to more than just me: Definition: A finite group is said to be ...