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Questions tagged [solvable-groups]

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

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Example of a finitely generated metabelian group whose Fitting subgroup is not nilpotent

It is known that the Fitting subgroup of a finitely generated polycyclic-by-finite group is nilpotent, but this statement is not true for the solvable group. It is clear that both Lamplighter groups ...
ghc1997's user avatar
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Let $G$ be a group of order $p^nq$ where $p$ and $q$ are distinct primes and suppose $q \nmid p^i-1$ for $1 \leq i \leq n-1$. Prove $G$ is solvable

This is an extension of this post. Let $G$ be a group of order $p^nq$ where $p$ and $q$ are distinct primes and suppose $q \nmid p^i-1$ for $1 \leq i \leq n-1$. Prove that $G$ is solvable. This can be ...
Grigor Hakobyan's user avatar
2 votes
0 answers
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Real forms of a solvable Lie algebra

Consider the Lie algebra $\mathfrak{r}_2=\mathfrak{aff}(\mathbb{C})$ of the group of affine maps of $\mathbb{C}$, and let $\mathfrak{g}=\mathfrak{r}_2 \oplus \mathfrak{r}_2$. I am interested in ...
JRojo's user avatar
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3 votes
1 answer
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Finite solvable Frattini-free group having a unique minimal normal subgroup N implies that N is the Fitting subgroup

This is exercise 6.1.6 of Kurzweil and Stellmacher. A restatement is: Let $G$ be a finite solvable group with $\Phi(G)=1$, and assume that $G$ has a unique minimal normal subgroup $N$. Then $N=F(G)$....
Alex Eustis's user avatar
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1 answer
56 views

Abelian groups that can be extended with non abelian solvable groups [closed]

Let $H$ be a non-trivial finite abelian group. Is it true (in some cases) that there exist a non-abelian solvable group $G$ and a normal abelian subgroup $K$ of $G$ such that $G/K$ isomorphic to $H$? ...
Rozina Ali's user avatar
1 vote
2 answers
46 views

$G$ is solvable and $H \trianglelefteq G$. Does $H$'s chain contain a tail end of $G$'s chain?

Let $G$ be a solvable group and let $H \lhd G$ be a normal subgroup of $G$. As $G$ is solvable, then $H$ is solvable. One equivalence of solvability (for $H$) is: there exists a chain of normal ...
Grigor Hakobyan's user avatar
4 votes
0 answers
53 views

Given a finitely generated metablian group $G$ with exponential growth, does the ratio of sizes of consecutive balls have a limit?

Suppose $G$ is a metabelian (hence solvable and amenable) group with exponential growth that is finitely generated by a symmetric set S. Given $k \geq 0$, let $B_k$ be the ball centred at the identity ...
ghc1997's user avatar
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Solvability by radicals, but you don't get to choose the roots

It is well-known that a polynomial equation $P(X)=0$ over a field $K$ is solvable by radicals if and only its Galois group is solvable. Here, "solvabile by radicals" is taken to mean that ...
Béranger Seguin's user avatar
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Example of 3-solvable group of large 3-length

I would like to construct examples of $3$-solvable groups with large $3$-length. That means that the Sylow $3$-subgroups need to be large in a sense. Is there any general construction of such examples?...
primer's user avatar
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Using Fredholm Alternative theorem to solve PDEs [closed]

I am trying to solve a coupled PDE system using the perturbation theorem, where parameter $\epsilon$ is small and all the variables are expanded in even powers of $\epsilon$. The set of equations and ...
mathlearner's user avatar
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1 answer
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What is the least degree polynomial whose Galois group is not solvable in the field of Bring rational numbers?

Let $\mathbb{Q}_{\mathrm{Br}}$ be the field of bring-jerrard rational numbers, this is, the field defined by the infinite algebraic extension $\mathbb{Q}_{\mathrm{Br}}=\mathbb{Q}(S)$, were $$S=\{\tau\...
Simón Flavio Ibañez's user avatar
3 votes
1 answer
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Equivalent Definition of Solvable Groups in Serre’s Book

In Serre's Linear Representations of Finite Groups, Section 8.2, the following claim is given, as equivalent to a definition of solvable groups. Solvable groups. One says that G is solvable if there ...
Daichi's user avatar
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Main idea of the proof of the Galois solvability criterion

For an oral exam I want to learn the proof of the Galois solvability criterion (Galois 1831). I have already seen the proof online, I tried to understand it but it is incredibly long. Could someone ...
Marco Di Giacomo's user avatar
2 votes
1 answer
85 views

Proof of a particular piece of Milnor-Wolf theorem

The Milnor-Wolf theorem says that a finitely generated solvable group that doesn't have exponential growth is virtually nilpotent. The proof I've seen is divided into two pieces: Prove that such a ...
Hempelicious's user avatar
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2 answers
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Is $SL_2(\mathbb{C})$ a solvable group?

Recently we showed in class that the group $SL_2(\mathbb{R}) $ is not solvable since the commutator subgroup of $SL_2(\mathbb{R}) $ is equal to $SL_2(\mathbb{R}) $, implying that the group $SL_2(\...
user007's user avatar
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3 votes
1 answer
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Let $G$ be a finite group and $f$ a non-trivial automorphism of $G$ such that for each $x\in G, f(x)=x$ or $f(x)=x^{-1}$. Prove that $G$ is solvable.

I've been working on a group theory problem but hit a roadblock, and I'd appreciate some guidance. The problem revolves around a finite group $G$ and a non-trivial automorphism $f$ in $G$. ...
123's user avatar
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2 votes
1 answer
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Is $p(x^n)$ solvable by radicals if $p(x)$ is solvable by radicals?

I have a question related to solvable by radicals polynomials. Is there any theorem or a result that assures me that if $p(x) \in \mathbb{Q}[x]$ is solvable by radicals then $p(x^n)$ is solvable by ...
user avatar
3 votes
2 answers
111 views

The solvability of a subgroup in $S_{11}$

Is the subgroup in $S_{11}$ generated by cycles $(1, 2, 3, 4, 5)$ and $(1, 6, 7, 8, 9, 10, 11)$ solvable? The cycles generating our subgroup have lengths $5$ and $7$, if multiplied, we get a cycle of ...
Quark's user avatar
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4 votes
1 answer
200 views

What is wrong with this proof that a polynomial is solvable by radicals?

We consider $$x^5 - x + 5 \in \mathbb{Q}[x].$$ I'm pretty sure that this polynomial is NOT solvable by radicals since its Galois group is isomorphic to $S_5$ (see my question from yesterday). However, ...
user avatar
1 vote
1 answer
119 views

How can I see if a polynomial is solvable by radicals or not?

We consider $$x^5 + 5x^4 + 10x^3 + 10x^2 + 5x - 2 \in \mathbb{Q}[x]$$ and $$x^5 - x + 5 \in \mathbb{Q}[x].$$ I want to see if they are solvable by radicals or not. I know that firstly I should ...
user avatar
2 votes
0 answers
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When does this Galois theorem "variation" holds?

Im reading about solvability by radicals in different books, more concretely in Fields and Galois Theory by Patrick Morandi and Introduction to abstract algebra by Benjamin Fine, etc. In the second ...
eldebarva's user avatar
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1 answer
48 views

Clarification over subnormal series of a group

I am reading through a lecture note on group theory and it says the following: Let $H \leq G$ be a subgroup of group $G$. If $G_n \trianglelefteq \cdots \trianglelefteq G_1 = G$ is a subnormal tower ...
love and light's user avatar
2 votes
1 answer
62 views

prove a finite nilpotent group has supersolvable series using upper center series only

Let $G$ be a finite nilpotent group. If G has a normal series $$\langle e\rangle=G_0\leq G_1\leq ... \leq G_n=G$$ such that $G_i\lhd G$ and $G_{i+1}/G_i$ is cyclic for all $i$, then G is called ...
N00BMaster's user avatar
5 votes
1 answer
256 views

Does Galois theory provide a method for solving solvable polynomials?

A polynomial $f(x)$ of degree larger than $5$ may not be solvable. But if we take, for example, $$ f(x)=x^8+x^6+x^4+2, $$ we can see by introducing a new variable $u=x^2$ (by observation) and it ...
CHWang's user avatar
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5 votes
1 answer
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Asymptotic density of certain class of finite groups (Solvable, Nilpotent, $p$-Group, etc).

I read that there is a conjecture that most groups are $2$-groups. This conjecture comes from the fact that by Higman-Sims asymptotic formula, $\#$ of $p$-group of order $p^k= p^{\frac{2}{27}k^3 + O(\...
Leon Kim's user avatar
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1 vote
1 answer
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presentation of a family of solvable groups

I am looking for a family of finite solvable groups (with infinite members) with given presentations. I am aware that for example dihedral groups, or dicyclic groups have well-known presentations. ...
Amin's user avatar
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0 votes
0 answers
125 views

Solvable extension is solvable by radicals

I have trouble understanding Serge Lang's proof that a solvable extension is solvable by radicals. More precisely, I cannot figure out why did he pick such a number. Proof. Assume that $E/k$ is ...
Degenerate D's user avatar
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1 answer
141 views

How to show that the normal subgroups in tower subgroups?

We say a tower $G=G_0\supseteq G_1 \supseteq G_2\dots \supseteq G_m$ is abelian if it is normal (i.e., each $G_{i+1}$ is normal in $G_i$) and if each factor group $G_i/G_{i+1}$ is abelian. I am ...
Hermi's user avatar
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0 votes
1 answer
68 views

Suppose $G$ is a solvable group and $|G|=n$, show that $G$ is isomorphic to a subgroup of group of upper triangular complex invertible matrices. [closed]

I've proved that all subgroups of upper triangular complex invertible matrices is solvable, but I find it too hard to show the inverse proposition.
ymx ddl's user avatar
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4 votes
0 answers
79 views

A different approach to proving a property of nilpotent injectors in solvable groups

Let $G$ be a finite solvable group. Call $J\subseteq G$ a nilpotent injector if it is a nilpotent subgroup that contains $\mathbf{F}(G)$, and that is maximal with this property (not properly contained ...
semisimpleton's user avatar
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0 answers
90 views

for $H \triangleleft G$, show that $H$ and $G/H$ solvable implies $G$ solvable [duplicate]

I have recently begun working my way through Serge Lang's Algebra, and on page 19 there is stated that for a group $G$ with a normal subgroup $H$, $G$ is solvable if and only if $H$ and $G/H$ are ...
paulina's user avatar
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0 votes
1 answer
57 views

Quotient group in solvable connected Lie group is isomorphic to real numbers or torus

I already know that when G is a connected solvable Hausdorff group there exists a sequence $$ G = G_0 \rhd G_1 \rhd ... \rhd G_R = (e)$$ where the $G_i$ are closed connected and $G_{i-1}/G_i$ is ...
diesmond's user avatar
2 votes
1 answer
124 views

Number of Sylow subgroups in $p$-solvable groups - Navarro article

This is a article which Gabriel Navarro wrote. I'm reading lemma 2.1. I see that "By standard arguments, recall that in any coprime action, if $q$ is a prime, then every $A$-invariant $q$-...
math_survivor's user avatar
1 vote
1 answer
160 views

Properties of the obvious action of $Aut(G)$ on $G$

Question: Let $G$ be a finite group. Let $Aut(G)$ be the group of automorphisims of $G$. Consider the group action $\phi:Aut(G)\times G\to G$ where $\phi(\sigma,g)=\sigma(g)$. Assume $G$ has exactly ...
confused's user avatar
  • 489
2 votes
1 answer
88 views

Solvable Groups and Derived Subgroups. Why does $D(G_i) \subset G_{i + 1} \implies D_i(G) \subset G_i$.

I'm working through the proof of the following theorem: $G$ is a solvable group $\Leftrightarrow$ There exists $n$ such that $D_n(G) = {1}$, where $D_i(G)$ is the $i$th derived subgroup of $G$. Proof: ...
Apollonius's user avatar
10 votes
1 answer
317 views

Group of order $1575$ is solvable

Let $G$ be a group of order $1575 = 3^25^27$. Show that $G$ is solvable. So far, my idea is to show that $n_7, n_5 $ or $n_3$ equals $1$, so that there exists a unique Sylow subgroup of order $7, 5^2$...
user2345678's user avatar
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1 vote
1 answer
71 views

Socle of a group and the Fitting subgroup of a solvable primitive group [closed]

I have been puzzled by this lately. And first, thank you to whoever takes the times to read through these questions. Consider a finite group $G$ which is solvable. Let $F(G)$ define the Fitting ...
Claudio Piedade's user avatar
0 votes
1 answer
666 views

showing that $S_n$ is not solvable for $n\ge 5$.

I am working on the following problem: Consider the symmetric group $S_n$ with $n\ge 5$. Prove the following: Show that any $3$-cycle is a commutator. Let $G$ be a subgroup of $S_n$ and let $H$ be a ...
SummerAtlas's user avatar
  • 1,042
2 votes
1 answer
106 views

Question on Hall's theorem for solvable groups

Let $G$ be a finite solvable group with $|G|=p_{1}^{\alpha _{1}}\cdot \cdot\cdot p_{r}^{\alpha _{r}}$ such that $p_{1}$,..., $p_{r}$ are distinct primes. By Hall's theorem, we have: If the number of ...
A R's user avatar
  • 115
3 votes
2 answers
145 views

When is the special linear group $SL_n(\Bbb F_q)$ over a finite field $\Bbb F_q$ solvable?

Recall that the special linear group $$SL_n(\Bbb F_q)=\{ A\in GL_n(\Bbb F_q)\mid \det(A)=1\},$$ where $\Bbb F_q$ is the field of $q$ elements for finite $q$. A group $G$ is solvable if the derived ...
Shaun's user avatar
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2 votes
1 answer
114 views

What does "proper" mean in this context?

Part a) of Problem 2C.1 of Isaacs' Finite Group Theory reads Show that every proper homomorphic image of an $N$-group is solvable. What does "proper" mean here? Please note that I'm not ...
Leandro Caniglia's user avatar
1 vote
1 answer
45 views

Corollary of the First Theorem of Hall for finite soluble groups

I have to prove the following statement: If $G$ is a finite soluble group and $N \trianglelefteq G$, then any $\pi$-Hall subgroup of $G/N$ is of the type $HN/N$ for some $H$ a $\pi$-Hall subgroup of $...
Sergio Ferrer's user avatar
2 votes
0 answers
97 views

A matrix group over commutative ring

Let $R$ be a commutative ring and denote by ${\rm GL}_n(R)$ the general linear group over $R$. Let $I_n$ denote the identity matrix of size $n$. Let $G$ be a subgroup of ${\rm GL}_n(R)$ such that ...
trivialquestions's user avatar
0 votes
2 answers
213 views

proof that if G is a solvable group then quotient of G is solvable

I am reading Stewart book on Galois theory, one of the theorem proves that if $G$ is solvable and $N$ is a normal subgroup of $G$, then the quotient of $G$, $G/N$ is solvable. I can't fully understand ...
Andrea's user avatar
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4 votes
1 answer
149 views

Solvable-by-finite groups

I am trying to prove this: Let $G$ be a finite-by-solvable group, i.e. $G$ has a normal subgroup $N$ that is finite with $G/N$ solvable. Prove that $G$ is solvable-by-finite, i.e., $G$ has a solvable ...
Milan Rashed's user avatar
2 votes
0 answers
49 views

Let $G$ be a group such that $N\unlhd G$ and $H \le G$ are both solvable. Show that $NH$ is solvable. [duplicate]

Since, $N \unlhd G$ and $H \le G$, from the second isomorphism theorem we know that $N \unlhd NH$, therefore $NH$ is solvable if $N$ and $NH/N$ are solvable. From our assumptions, $N$ is solvable, ...
NoCap's user avatar
  • 55
3 votes
2 answers
229 views

Solvability of quintics with complex coefficients?

I was trying to explain Galois theory to a non-specialist and was given a question I couldn't solve to my satisfaction, apologies if I have missed something obvious. By Galois theory, a polynomial $f(...
Uzai's user avatar
  • 497
1 vote
1 answer
158 views

If $G_f\cong D_5$ for $f(x)=x^5+ax+b\in \mathbb{Q}[x]$, show that these criteria are true.

As a followup to this post, I was wondering if the converse of this statement was true. That is, if $G_f$ (the galois group of $f(x)=x^5+ax+b$ for $a,b\in\mathbb{Q}$) is isomorphic to $D_5$ (the ...
IAAW's user avatar
  • 1,516
2 votes
1 answer
132 views

Affine type primitive group

By the definition it looks like 'Every solvable primitive groups are of affine type primitive groups only.' Is the converse true ? i.e. Is it true that every affine primitive permutation groups are ...
Jins's user avatar
  • 564
1 vote
1 answer
422 views

A group is solvable if it has only one $p$-Sylow subgroup for each $p$

The problem is to prove that a group is solvable if it has just one $p$-Sylow subgroup for each prime $p$ dividing its order. My solution: If$|G|=p_1^{\alpha_1}p_2^{\alpha_2}\dots p_n^{\alpha_n}$ ...
stranger's user avatar
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