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

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2
votes
2answers
28 views

Some solvable Lie algebra but not nilpotent

Can someone provide two concrete examples the Lie algebra which is solvable, but not nilpotent? -- And further explain the subtle differences between the solvable Lie algebra and the ...
1
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1answer
49 views

Prove $G$ is cyclic if for every $n\in \Bbb{N}$, there are at most $n$ solutions to the equation $x^n=e$. [duplicate]

Let $G$ be a finite group. Prove G is cyclic if for every $n\in \Bbb{N}$, there are at most $n$ solutions to the equation $x^n=e$. I am stuck with this :( Would appreciate your help.
2
votes
1answer
61 views

What is set builder of $\langle H, K \rangle$?

I am looking left and right for a lemma to solve a problem on solvable group here, and I think I have found one under commutator group: For any two subgroups $H, K$ of $G$, the $[H, K]$ is a ...
1
vote
2answers
128 views

Reconciling Different Definitions of Solvable Group

Looks like my class note defines solvable group differently from others: A finite group $G$ is called solvable if $H' \neq H$ for each subgroup $H$ of $G$ different from $\{1\}$, where $H'$ is ...
4
votes
1answer
58 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 ...
0
votes
0answers
24 views

$S/R_u(S)$ for solvable groups

Consider a solvable linear algebraic group $S$ over a field $F$ with $char(F)=0$. Let $R_u(S)$ be the unipotent radical of $S$. If $S$ were connected, then $S/R_u(S)$ would be a torus. In ...
6
votes
1answer
70 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 ...
2
votes
2answers
72 views

Are polynomials over $\mathbb{R}$ solvable by radicals?

I know that if $f$ is a polynomial over a subfield $F$ of $\mathbb{C}$ and $f$ is solvable by radicals, then the Galois group of $f$ over $F$ is solvable. I've also seen many applications of this fact ...
2
votes
1answer
23 views

Understanding the definition of solvable by radicals.

I am currently studying the third edition of Ian Stewart's book "Galois Theory". In the book, solvability of a polynomial by radicals is defined as follows: Let $f$ be a polynomial over a subfield ...
3
votes
1answer
22 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? ...
0
votes
0answers
20 views

Is this a valid way to extend the proof of the insolubility of the quintic?

I'm just musing here, this is only (barely even) a half-formed idea, but I'm just wondering if it's at all a valid train of thought. The proof I read of the insolubility of the quintic polynomial ...
1
vote
1answer
49 views

Minimal non-countable Groups

I was thinking the following thing: Is there an uncountable group whose all proper subgroups are countable which is also for instance locally soluble? I've found some example of minimal ...
0
votes
2answers
37 views

Intersection of composition factors

For $G=HK$, and $K= K_1\cap K_2$, both normal in $G$, if $G/K_1$ and $G/K_2$ are solvable, show that $G/K$ is solvable. By the Third Isomorphism Theorem, $$\frac{G/K}{K_i/K}\cong \frac{G}{K_i}$$ ...
1
vote
0answers
29 views

Generating a group by its $q$-elements.

Let $G=PQ$ be a solvable group where $P$ and $Q$ are $p$-subgroup and $q$-subgroup of $G$ respectively. Also suppose that $Q$ is not normal in $G$. Is it true that the group generated by all ...
1
vote
0answers
30 views

If all sylow subgroups are cyclic, prove that G is solvable

I came across a statement which I am unable to prove by myself that if $G$ is a finite group then if all its sylow subgroups are cyclic, prove that G is solvable. If it has been asked before please ...
0
votes
1answer
48 views

Nilpotent torsion-free Groups with a fixed Soluble length

Let $s$ be a natural number. Is it possible to find, for each $n$ natural number greater than some arbitrary constant, a torsion-free group whose nilpotency class is $n$ and soluble length is $s$?
0
votes
2answers
19 views

Isomorphism type of two quotient groups of $G=\mathbb{Z}^\times_{16}$

I'm supposed to determine the isomorphism type of $G/\langle 15\rangle $ and $G/\langle 9 \rangle$. I've determined the order of both subgroups ($\langle 15\rangle$ and $\langle 9\rangle$), and it is ...
1
vote
0answers
37 views

Solvability by radicals of Polynomials defined by a recurrence relation

I want to determine the smallest integer $m$ such that the polynomial $P_{n}(x)$, $n\geq m$, given by : $$\left \lbrace \begin{array}{l} P_{n+1}(x) = P_n(x) (x-n-1) + \prod\limits_{i = 0}^n x-i\\ ...
1
vote
0answers
42 views

Lucido's three prime lemma

I am looking for proof of this statement I encountered in a paper. $\textbf{(Lucido’s Three Primes Lemma)}$- Let $G$ be a finite solvable group. If $p, q, r $ are distinct primes dividing |$G$|, ...
0
votes
0answers
100 views

finite solvable group with a certain property

Let $G$ be a finite solvable group and for each proper normal subgroup $N$ of $G$, $\frac{G}{G^{\prime}N}\cong \Bbb{Z}_p\times\Bbb{Z}_p$ or $\Bbb{Z}_{p^n}$, where $n\geq 1$, $p$ is a prime number ...
2
votes
1answer
40 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 ...
1
vote
1answer
72 views

Hall's Subgroup Theorem

I am currently looking into the extension of Sylow's theorems, namely through Hall-$\pi$-subgroups and Hall's Theorem. I currently have the theorem as; Let $G$ be a finite solvable group a $\pi$ be ...
1
vote
0answers
57 views

Supersolvable groups of order $pq^m$

According to Burnside's classification in his book "Theory of groups of finite order", one of the types of non-abelian groups of order $pq^2$ ($p$ and $q$ are distinct primes), has the presentation ...
0
votes
0answers
23 views

Lie algebra: If ad(g) is solvable then g solvable?

I'm trying to prove that if the image of the adjoint representation of a Lie algebra g is solvable then g is solvable, ie, if for some n (ad(g))^(n) = 0 then there exists a m such that g^(m) = 0 My ...
1
vote
1answer
112 views

Solvable implies quotient group is solvable: Proof check.

I'd like to check the veracity of my proof. I've seen several proofs using different methods (some I'm allowed to use with lots of element-pushing and others using ideas I'm not allowed), but none ...
4
votes
1answer
72 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?
2
votes
1answer
65 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 ...
8
votes
1answer
70 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 ...
3
votes
0answers
38 views

Determining if a given equation is solvable given a set of ultra-radicals

So suppose someone is armed with the tools of standard arithmetic, exponents (and of course that comes along with roots) AS WELL AS a set of inverses for some polynomials which are not solvable using ...
4
votes
0answers
70 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 ...
6
votes
0answers
62 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 ...
3
votes
4answers
196 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 ...
6
votes
3answers
339 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 ...
1
vote
0answers
23 views

series of subgroups of the solvable group $Gal(x^6-7) $.

I have to solve the following question: Let $p(x) = x^6-7 \in \mathbb{Q}[x]$, then $Gal(p(x)) $ is a solvable group (the splitting field of $p(x)$ is contained in a radical extension). Give a series ...
0
votes
2answers
70 views

Minimal non solvable group is simple

I suppose to prove in my homework that Every minimal non solvable group is simple. I can't find the way. Thank you.
0
votes
1answer
35 views

Is it sufficient that G is solvable?

Given a normal subgroup H of G, H and G/H are solvable. Then is G solvable? I know the converse is true... but I have no idea for our statement.
4
votes
1answer
53 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 ...
1
vote
1answer
44 views

Nilpotent group with torsion divisible abelian quotient

Just want to make sure this is true: If $G$ is a nilpotent group such that $G/[G,G]$ is a torsion divisible abelian group (like $\mathbb{Q}/\mathbb{Z}_{(p)}$), then $G$ is abelian. I get that ...
1
vote
1answer
205 views

Homomorphic Image

If $N$ is any normal subgroup of $G$, then the factor group $G/N$ is abelian if and only if $G' \subseteq N$. In the proof I don't understand why $G/N$ is the homomorphic image of $G/G'$ $G\subseteq ...
1
vote
2answers
60 views

nilpotent group implies solvable group

Can someone please do a simple proof of this: If a group, G, is nilpotent then it is solvable. I'm pretty bad at math and am just trying to figure this out. Thank you very much!
0
votes
1answer
56 views

Subgroup of solvable group is solvable (not using tower definition)

I'm studying for an exam, and I'm having problems proving that subgroups of solvable groups are solvable. I want to use this definition of solvability: A group $G$ is solvable if and only if ...
0
votes
1answer
24 views

Proof with solvable groups

I need help with this proof please: prove: if $G$ is a solvable group then $G$ cross $G$ is a solvable group.
0
votes
1answer
30 views

A group of order less than $60$

I'm given a theorem (call it Theorem 1) which states that $G$ is solvable iff there exists a chain of normal subgroups $\{e\} = G_0 \unlhd \ldots \unlhd G_n = G$ such that $G_i / G_{i-1}$ is abelian ...
3
votes
2answers
85 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
1answer
53 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 ...
0
votes
1answer
77 views

Prove that $F(\sqrt{\alpha}) = F(\sqrt{\beta})$

Let $F$ be a field of characteristic $\neq 2$. State and Prove a necessary and sufficient condition on $\alpha, \beta\in F$ so that $F(\sqrt{\alpha})=F(\sqrt{\beta})$.
2
votes
1answer
64 views

On the Hall Theorem

I went to search info about it in books (as Isaac), this site, wikipedia. But no one says nothing about my (maybe silly) doubt. The theorem states that, given a finite and solvable group $G$, and an ...
2
votes
0answers
109 views

Are the groups of order $p^3q^5$ solvable?

Problem: Let $G$ be a group of order $p^3q^5$ where $p$ and $q$ are two distinct prime numbers. Is $G$ solvable? If it is, how to justify it (without using the Burnside's Theorem [wiki])? ...
0
votes
1answer
42 views

How to prove that $G/(H \cap K)$ is solvable given $H \triangleleft G, K \triangleleft G$ and both $G/H$ and $G/K$ are solvable using Abel series?

The problem is as follows: Let $G$ be a group. $H \triangleleft G, K \triangleleft G$. To prove that: both $G/H$ and $G/K$ are solvable $\iff$ $G/(H \cap K)$ is solvable. This proposition is ...
1
vote
1answer
67 views

An alternative definition of a solvable group

I'm putting together an outline for a paper on Galois theory. There are a few equivalent definitions of a solvable group, and I need to make sure that the one I'd like to use works, or more ...