Questions tagged [solvable-groups]
For questions on solvable groups, their properties, and structure.
520
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Does every normal subgroup of a finite solvable group appears in some subnormal series whose factor groups are all abelian? [duplicate]
Let $G$ a finite solvable group.
Does every normal subgroup of $G$ appears in some subnormal series whose factor groups are all abelian?
Any comment is welcome.
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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 ...
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1
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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 ...
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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 ...
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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 ...
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1
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59
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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(\...
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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. ...
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55
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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 ...
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110
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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 ...
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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.
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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 ...
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61
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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 ...
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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 ...
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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 $q$-...
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1
answer
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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 ...
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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:
...
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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 $7, 5^2$...
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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 ...
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1
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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 ...
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1
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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145
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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 ...
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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, ...
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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(...
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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 ...
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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 ...
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1
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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$|G|=p_1^{\alpha_1}p_2^{\alpha_2}\dots p_n^{\alpha_n}$ ...
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1
answer
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Variations on Hall Theorem for solvability in Dummit and Foote
In "Abstract Algebra" by Dummit and Foote p.105 there is a theorem:
The finite group $G$ is solvable if and only if for every divisor $n$
of $|G|$ such that $(n, \frac{|G|}{n}) = 1$, $G$ ...
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0
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On a notion involving coprime commutators in finite groups
In a finite group $G$, a commutator $[x,y]$ is called a coprime commutator if $(|x|,|y|)=1$. In an article of Shumyatsky, he defines a certain notion involving these coprime commutators which he in ...
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Let $G$ be generated by $\begin{pmatrix}1&p\\ 0&1\end{pmatrix},\begin{pmatrix}1&0\\ p&1\end{pmatrix}$ for prime $p$. Is $G$ is solvable? A proof.
Here is a problem determining the solvability of the given group.
Let $G$ be a subgroup of $\text{GL}_2(\mathbb{R})$ generated by two elements
$$
\begin{pmatrix}1 & p \\ 0 & 1 \end{pmatrix}, \...
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Does this lemma still holds when replacing 'Galois' by ‘normal’?
All of the field extensions involved are assumed to be finite.
This lemma is from Isaacs' Algebra, serving as a key step toward the establishment of Galois' criterion of solvability. The proof of the ...
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A question about the proof that a polynomial is solvable by radicals iff Galois group is solvable (Theorem 14.7.39 of Dummit and Foote)
My question is about the following theorem:
A polynomial $f(x)$ can be solved by radicals if and only if its Galois group is solvable.
The difficulty arises in the last assertion in this paragraph:
...
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Fitting length of Dihedral groups
Let $G$ be a dihedral group of order $2n$ where $n\geq 1$, denoted by $D_n$. We know that $G$ is nilpotent if and only if $n=2^i$ for all $i\geq 1$, a proof of this you can check in the below link 1. ...
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Group of order $630$ is solvable
I tried using the Sylow theorems. $|G|=630=2\cdot 3^2 \cdot 5 \cdot 7$.
Denot the number of $p$-Sylow groups by $k_p$ then:
$k_2 \in \{1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 315\}$
$k_3 \in \{1, 10, 70\}$
...
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Every ordinal is the derived length of a group
Let $G$ be a group. Define inductively the derived series as follows:
$$G^{(0)}=G\\\\
G^{(\lambda)}=\bigcap_{\alpha<\lambda}[G^{(\alpha)},G^{(\alpha)}]$$
Let $\text{sol}(G)=\{\min \alpha\in \text{...
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Effective degree bound for solvability by radicals
Let $P\in{\mathbb Q}[X]$ be an irreducible polynomial of degree $n\geq 3$, and let $\mathbb L$ be the decomposition field of $P$. Denote the Galois group of the extension ${\mathbb L}:{\mathbb Q}$ by $...
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2
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Reference or idea of a proof (on A-groups)
There is the following result on wiki:
The Fitting subgroup of a solvable $A$-group is equal to the direct product of the centers of the terms of the derived series.
An $A$-group is a (finite) group ...
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1
answer
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Doubt in Lang's proof of solvable extensions forming a distinguished class of extensions
The context for this question is the same as that is described in How to see that $M$ is Galois over $k$ in Lang's proof that solvable extensions are a distinguished class? (Prop. VI.7.1, *Algebra*...
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1
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Solvable normal subgroup but the corresponding quotient group is not solvable [closed]
Let $G$ be a group and $N$ be a normal subgroup of $G$.
It is a well known fact that: $G$ is solvable iff $N$ and $G/N$ are solvable.
I wonder the following:
Could you give an example of a group $G$ ...
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0
answers
35
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Problem (12.4) in Isaacs Character theory of finite groups
I am trying to solve problem (12.4) in Isaacs' Character theory of finite groups.
The problem is to show that if $G$ is non-abelian solvable and has no non-abelian factor group of prime power order, ...
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0
answers
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What is the quotient group $G/R(G)$?
The maximal solvable normal subgroup of $G$ is called the radical subgroup of $G$, and it is denoted by $Rad(G)$ of $R(G)$.
Question: Is the quotient group $G/R(G)$ been classified (i.e., it always ...
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1
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1
answer
64
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Why $\mathscr{D}^k\mathfrak{g}$ is an ideal in $\mathfrak{g}$?
We have defined the derived series $\{\mathscr{D}^k\mathfrak{g}\}$ inductively by
$$\mathscr{D}^1\mathfrak{g}=[\mathfrak{g},\mathfrak{g}],\ \mathscr{D}^k\mathfrak{g}=[\mathscr{D}^{k-1}\mathfrak{g},\...
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1
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1
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Profinite group with $2$-generated closed prosoluble subgroups
I'm working in a problem and in the order to solve it, I'm stuck on a claim.
Suppose that each $2$-generated closed subgroup of a profinite group $G$ is prosoluble. Is it true that $G$ is prosoluble?
...
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154
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A solvable group is characteristically simple if and only if is abelian elementary
Let $G$ a finite group. I know that
$G$ is solvable $\iff$ every principal factor is abelian elementary
I already showed that if $G$ is a abelian elementary group then $G$ is characteristically ...
0
votes
1
answer
120
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When a solvable group is supersolvable?
I am thinking about the following question:
We know if group $G$ has a normal subgroup such that $N$ and $G/N$ are solvable, then $G$ is solvable. But this is not true for the supersolvable groups.
...
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1
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73
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Doubt in proof of Herstein Theorem (Solvable Groups) [duplicate]
I'm trying to prove this result due to Herstein (1958):
If $G$ is a finite group and $A \le G$ is a maximal subgroup. If $A$ is abelian, then G is solvable.
Since $A$ is maximal, we have only 2 ...