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

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2
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1answer
23 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
38 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
47 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
17 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
36 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
68 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
60 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 ...
7
votes
1answer
52 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
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0answers
34 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
59 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
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0answers
52 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
134 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
138 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
66 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
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1answer
31 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
50 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
41 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
81 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
49 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
43 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
21 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
26 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
79 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
48 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
73 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
62 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
88 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
36 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
62 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 ...
6
votes
1answer
87 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 $G/[G,G]$ ...
0
votes
1answer
44 views

A question on solvable groups.

Let's suppose $S$ is a group and can be written as $S= G_1\times G_2\times\cdots\times G_K$; here each $G_i$ is a group of prime power order. And $o(S)= n.$ Is this $S$ solvable group? If yes how will ...
4
votes
1answer
65 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 ...
0
votes
0answers
41 views

Alt(6) is a subgroup of Alt(7)

How can it be shown that $Alt(6)$ is a subgroup of $Alt(7)$? Can we also prove the fact that $Alt(7)$ is not solvable?
1
vote
1answer
83 views

Proof for: semidirect product of solvable groups is solvable

Do you know the proof for the following statement or where I can find it? Semidirect product of solvable groups is solvable? I thought that this property is so ...
7
votes
1answer
237 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 $G$ ...
2
votes
1answer
41 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 ...
1
vote
0answers
29 views

Non-solvable, closed subgroups of $\mathrm{PSL}(2,\mathbb{R})$

It is mentioned here that non-solvable closed subgroups of $\mathrm{PSL}(2,\mathbb{R})$ are either the entire space or discrete. My question is this: Is there any easy proof of this, or do any of you ...
7
votes
4answers
506 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 ...
5
votes
2answers
322 views

Why is a group of order 135 nilpotent?

Why does order 135 imply nilpotent?
0
votes
0answers
62 views

Express the exact root using radicals.

In Galois Theory, when a polynomial is not solvable by radicals, does that imply that the real solutions cannot be expressed exactly using the operators $+,-,\times,/$ and/or radicals?
2
votes
0answers
187 views

Maximal subgroups of direct product of solvable groups

Let $G_1$ and $G_2$ be finite solvable groups and $M$ be a maximal subgroup of $G=G_1\times G_2$ show that one of the following holds: $$ G_1\times\{e\}\leq M,\ \{e\}\times G_2\leq M,\ M \unlhd G.$$ ...
0
votes
2answers
53 views

If the product $x_1x_2…x_n=1$, prove that each $x_i=1$.

Let $G$ be a finite solvable group and let $x_1,x_2,...,x_n$ be elements of $G$ of pairwise relatively prime orders. If the product $x_1x_2...x_n=1$, prove that each $x_i=1$. I have no idea. Tell me ...
1
vote
1answer
360 views

Prove that if a group is nilpotent , Than it's quotient in its frattini subgroup is abelian.

I know that : 1) Nilpotent group is solvable. 2) Subgroup of a solvable group is solvable. 3) Solvable and simple group is abelian. Now I should use these facts to prove it.
2
votes
1answer
125 views

Show that the minimal polynomial of every element in $K=\mathbb{Q}(\zeta)$ is solvable by radicals, where $\zeta$ is a primitive 9th root of unity.

I have found the minimal polynomial if $\zeta$ over $\mathbb{Q}$ is $x^{6}+x^{3}+1$. $\mathbb{Q}(\zeta)\colon\mathbb{Q}$ is a normal and separable extension so ...
1
vote
1answer
52 views

Standard reference for Kaluzhnin's theorem

Kaluzhnin's theorem says that if $G$ is a group and $H \leq \operatorname{Aut}G$ acts trivially on each step of the normal series $1 = G_0 \leq G_1 \leq \ldots \leq G_n = G$, then $H$ has nilpotency ...
4
votes
1answer
160 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 ...
2
votes
2answers
81 views

Finite extensions of nilpotent groups

Let $G$ be a torsion free, nilpotent group and let $H$ be a normal subgroup of $G$ such that $G/H$ is a finite cyclic factor group. It is true that if $H$ is nilpotent of class $c$ then also $G$ ...
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votes
2answers
135 views

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. [closed]

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. Help me some hints.
3
votes
3answers
112 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.