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

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4
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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}...
2
votes
1answer
80 views

No group has commutator isomorphic to S_4

I'm wondering about an alternative solution to the following question from Dummit and Foote: Show that no group has commutator subgroup isomorphic to $S_4$. The hint says to use the previous ...
1
vote
1answer
49 views

Nilpotency class of dihedral group $D_{16}$

Using only upper central series, find the degree of nilpotency of the dihedral group $G = D_{16}$. (Answer: $3$) So I need to find the non-negative integer $c$ such that $Z^c(G) = G$, where $Z^c(G)$ ...
1
vote
1answer
47 views

Sylow $q-$radical subgroup of a solvable group

Let ‎‎$G‎‎$ ‎be a‎ ‎finite ‎solvable ‎group ‎of ‎order ‎‎$p^2q^2‎‎$‎, where ‎$p>q‎$ ‎and ‎$‎q\nmid‎‎ ‎p-1‎$‎‎. ‎Let‎ ‎‎$G‎‎$ ‎has ‎the ‎following ‎presentation‎:‎ $‎‎‎\langle a‎ , ‎b‎ ,‎c \vert a^...
0
votes
1answer
32 views

On solvable group and normalizer

" Let $G$ be finite group, $H$ is a subgroup of $G$, and $P$ is a Sylow-p Subgroup of $G$. If $N_G(P) \leq H$, show that $N_G(H)=H.$ " This problem appears in Martin Isaacs book under the chapter ...
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,\...
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 ...
3
votes
0answers
96 views

The groups with nilpotent hall $p'$ subgroup.

Theorem $1$(Burnside): A simple nonabelian finite group can not have a conjugacy classes with prime power elemets. Theorem $2$: A group of order $p^nq^m$ is solvable. Theorem $1$ depends on ...
3
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0answers
48 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 ...
3
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0answers
234 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: $$ G_1\times\{e\}\leq M,\ \{e\}\times G_2\leq M,\ M \unlhd G.$$ ...
2
votes
0answers
31 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 ...
2
votes
0answers
160 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])? ...
2
votes
0answers
209 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 ...
1
vote
0answers
25 views

Doubly transitive and solvable

I wonder if there are subgroup of $S_n$ that are solvable and doubly transitive for $n\geq 5$. Can someone find an example or prove that they don't exist?
1
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0answers
52 views

Which unitary groups are solvable?

I know that if $\zeta$ is a $n^{th}$ primitive root of the unity, we have :$$Gal(\mathbb{Q}[\zeta ]|\mathbb{Q})\cong U(n)$$ Where $U(n)$ is the group of units. I was wondering, if there were some ...
1
vote
0answers
21 views

If a group $G$ is solvable, is the subnormal series (the chain of the normal subgroups) unique?

If a group $G$ is solvable, is the subnormal series (the chain of the normal subgroups) unique? Or what type of solvable groups have unique subnormal series?
1
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0answers
33 views

Proving that a finite group $G$ is solvable iff for every divisor $n$ of $|G|$ such that $(n, |G|/n) = 1$, $G$ has a subgroup of order $n$.

I found this theorem in Dummit and Foote, and there was no proof of it there. It looks difficult to prove and I also could not find any resources online to help me out with this theorem. So here is ...
1
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0answers
33 views

Proof of solvability of B2 group

I am trying to understand the following proof of solvability for the group $B_2$. Let $B_2 = \{\begin{pmatrix}a&b\\&d\end{pmatrix}:ad\in\Re^x, b\in\Re\}$ Let $U_2 = \{\begin{pmatrix}1&b\\...
1
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0answers
26 views

Quotient is isomorphic exercise

Suppose $G$ is solvable, $N \vartriangleleft G$. Let $f \in Hom(G,H)$. We have a normal series $\{e\}=G_0 \vartriangleleft G_1 \vartriangleleft ... \vartriangleleft G_n = G$ with $G_{i+1}/G_i$ ...
1
vote
0answers
12 views

What tools are there for specific cases of Feit-Thompson?

Suppose I wanted to show that, for a specific odd order, any finite group of that order is solvable. What tools are available to solve such a question? I'm asking since I was thinking about Feit-...
1
vote
0answers
32 views

$Z(\mathcal{L}^{(n)}) \subset Z(\mathcal{L})$ for solvable Lie algebras?

$X$ Banach space. $\mathcal{L} \in B(X) $ is solvable Lie Algebra. Then for some n, $\mathcal{L} \supset \mathcal{L}^{(1)}=[\mathcal{L},\mathcal{L}] \supset \mathcal{L}^{(2)}=[\mathcal{L}^{(1)},\...
1
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0answers
31 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 $q$-...
1
vote
0answers
58 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 ...
1
vote
0answers
47 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\\ P_0(...
1
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0answers
55 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 |$G$|, ...
1
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0answers
73 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 ...
1
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0answers
36 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 ...
1
vote
0answers
89 views

$G$ be a non-nilpotent and supersoluble finite group.

Let $G$ be a non-nilpotent and supersoluble finite group. $G=Q\ltimes P$ and $|Q: C_{Q}(P)|=q$ is prime, Where $P$ is a group of prime order $p\neq q$. $Q$ is noncyclic group of order $|Q|=q^{\...
1
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0answers
69 views

$G$ be a $p$-supersoluble group. and $O_{p'}(G ) = 1$ then $G$ is supersoluble

Let $G$ be a $p$-supersoluble group. If $O_{p'}(G ) = 1$ then $G$ is supersoluble.
1
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0answers
102 views

Supersolvable group and nilpotent maximal subgroup

$G$ is a supersolvable group, $|G|=pq^{b}$ ($p$ and $q$ are different prime numbers). $Q$ is a $q$-Sylow subgroup of $G$, $Q=\langle x\rangle$ and $\operatorname{Cor}_{G}(Q)=\langle x^{q}\rangle$. If $...
0
votes
0answers
56 views

About correct meaning of radical solution to polynomials

Suppose that we have the equation $(1)$ $x^2 = a$ Whose root is $ x = \mp \sqrt{a}$. This is the "radical solution" of the equation. Suppose that we have $\sqrt[3]{m +\sqrt{n}}+ \sqrt[3]{m -\sqrt{...
0
votes
0answers
33 views

Can the number of non-solvable groups of a given order be easily determined?

It can be extremely difficult to find the number of groups of a given order. But if we only want to find the number of non-solvable groups of a given order, is there an easy algorithm doing the job ?...
0
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0answers
21 views

A finite group is solvable iff the simple factors in a decomposition sequence are abelian

Show that a finite group $G$ is solvable group (in the sense there exists an $n$ such that $G^{(n)}=1$) if and only the simples factors in a decomposition sequence of $G$ are all abelians. I'm not ...
0
votes
0answers
10 views

Proving that $\mathbb{Q}^{\text{sol}}$ isn't hilbertiean

I was given the following two definitions: A field $F$ is hilbertiean if for every $f(x,y)\in K[x,y]$ which is irreducible over $K(x)[y]$ there are infinite $a\in K$ s.t $f(a,y)$ is ...
0
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0answers
25 views

Showing that $ S_{4} $ isn't $ 2 $-supersoluble

I want show $ S_{4} $ isn't $ 2 $-supersoluble . $ S_{4} $ has a chief series. $ 1 < V_{4} < A_{4} < S_{4} $ is the chief series of $ S_{4} $. It is true said that $ S_{4} $ isn't $ 2 $-...
0
votes
0answers
28 views

If $ G $ is a finite soluble group and satisfies the permutizer condition, then for any odd prime $ p $, $ G $ is $ p $-supersoluble.

If $ G $ is a finite soluble group and satisfies the permutizer condition, then for any odd prime $ p $, $ G $ is $ p $-supersoluble. It is for me problem that why $ G $ can't $ 2 $-supersoluble group....
0
votes
0answers
42 views

find the factor groups of the p group - G and prove that G is solvable , where $|G|= p^a$ , p is prime

G is a p-group - $|G| = p^a$ p is prime i need to find the factor groups of G and prove that it is solvable. what i tried - EDITED: after watching the comments and investigating I know every ...
0
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0answers
34 views

Let $ K/L $ be a chief factor of $ G $ and $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent

Let $ G $ is soluble group with $ \Phi(G) = 1 $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Let $ K/L $ be a chief factor of $ G $ and $ M $ be the smallest normal ...
0
votes
0answers
29 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 ...
0
votes
0answers
189 views

Every group of order 1000 is solvable

Here is how far I am: $1000 = 5^{3}\cdot 2^{3}.$ Number of $5$-Sylows is $1$ mod $5$ and has to divide $8,$ so its $1$ making it normal. Lets call it $F.$ Then $G/F$ has order $2^{3}$ and is thus ...
0
votes
0answers
81 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 ...