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

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30
votes
1answer
431 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 ...
13
votes
1answer
316 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 ...
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 ...
5
votes
1answer
513 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 ...
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 ...
0
votes
0answers
57 views

Solvable group with ACC and DCC must be finite

Could some one please send me along a right path? I am unsure how to fit the "solvability" condition in this: If a solvable group's poset of subgroups satisfies both the ascending and descending ...
0
votes
2answers
84 views

Finite Group soluble $\iff G^{(k)}=\{e\}$ for some $k$

I am trying to understand the proof of the following proposition: A finite group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$, where $G^{(0)}=G$, $G^{(i+1)}=[G^i G^i]$ is the derived series. ...
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 ...
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?
4
votes
1answer
123 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 ...
2
votes
0answers
223 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: $$ G_1\times\{e\}\leq M,\ \{e\}\times G_2\leq M,\ M \unlhd G.$$ ...
7
votes
1answer
272 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 $G$ ...
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 ...
2
votes
0answers
130 views

Class of $p$-supersolvable group is saturated formation.

I want to show that the class of $p$-supersolvable groups is a saturated formation. I only have the definition of $p$-supersolvable. What do I do? A group is $p$-supersolvable iff every chief factor ...
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 ...
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 ...
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 ...
0
votes
1answer
78 views

$G$ be a non-nilpotent and supersoluble and 2-maximal subgroup of G permutes with all 3-maximal subgroup of G

Let $G$ be a non-nilpotent and supersoluble finite group. $G=Q\ltimes P$, Where $P$ is a group of order $p^{2}$($p$ is prime), all maximal subgroups of $P$ are normal in $G$, $Q=\langle a\rangle$ is ...