Questions tagged [solvable-groups]

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

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38 views

Applications of Tits' alternative in number theory

I have recently studying Tits' alternative. The theorem statement goes like the following: Tits' alternative: Let $G$ be any finitely generated linear group over a field. Then one of the following is ...
3
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1answer
74 views

Prove that if $G$ is a finite group in which every proper subgroup is nilpotent, then $G$ is solvable.

Prove that if $G$ is a finite group in which every proper subgroup is nilpotent, then $G$ is solvable. (Hint: Show that a minimal counterexample is simple. Let $M$ and $N$ be distinct maximal ...
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2answers
92 views

Let $G$ be a group with order $105 = 3 \cdot 5 \cdot 7$

(a) Prove that a Sylow $7$-subgroup of $G$ is normal (b) Prove that $G$ is Solvable Can anyone please tell me if I am correct? (a) For the sake of contradiction suppose $G$ dose not have a normal ...
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A short question about Hall $\pi$ subgroups

Let $\pi$ be a set of primes. Our definition for Hall $\pi$-subgroups states: A Hall $\pi$-subgroup of $G$ is a subgroup $H$ where $|H|$ is product of elements of $\pi$ and $|G:H|$ is product of $P\...
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2answers
33 views

G is solvable iff factors have prime order [duplicate]

A group $G$ is said to be solvable if, and only if, there exists a subnormal series of subgroups $\{e\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G$ such that each factor $\...
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1answer
43 views

On the number of invariant Sylow subgroups under coprime action -Antonio Beltrán, Changguo Shao

I'm reading the papers of Antonio Beltrán, Changguo Shao. The article is On the number of invariant Sylow subgroups under coprime action: https://www.researchgate.net/publication/...
2
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1answer
58 views

Is the $S_4\times G$ solvable group?

We have the following claim : The group $G$ is solvable iff $S_4\times G$ is solvable. If we consider that $S_4\times G$ is solvable we have that $1\times G\leq S_4\times G$ is solvable as a subgroup ...
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1answer
54 views

group of order $p^nq^m$ is solvable

I know Burnside theorem says this is true I wonder can i go easy way if there is an additional condition: $p<q$ and the order of $[p] \in \mathbb{Z}_q^{\times}$ is larger than $n$
2
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1answer
27 views

Heisenberg group is nilpotent

$F$ is a field and $H(F)$ is the Heisenberg group over $F$. Is it nilpotent? Is it solvable? I did all the math and I found that the commutator subgroup is in the center $Z(H(F))$, so $H(F)/Z(H(F))$ ...
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1answer
46 views

$G$ is solvable implies there exists a chain of normal subgroups such that adjacent quotients are cyclic

This is one part of me trying to solve exercise 3.4.8 in D&F Abstract Algebra. In particular I am proving (a) implies (b), and am frustrated with the method I found because it involves nested ...
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2answers
177 views

Index of subgroups in a finite solvable group, with trivial Frattini subgroup (Exercise 3B.12 from Finite Group Theory, by M. Isaacs)

Let G be a finite solvable group, and assume that $\Phi(G) = 1$ where $\Phi(G)$ denotes the Frattini subgroup of G. Let M be a maximal subgroup of G, and suppose that $H \subseteq M$. Show that $G$ ...
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1answer
67 views

Group of order $q^3p^3$, where $p,q$ are twin primes greater than $10$, is solvable

Let $q>p>10$ be twin primes, i.e., $q=p+2$. Show that every group of order $q^3p^3$ is solvable. This should be proven without using Burnside's theorem. Looking at the Sylow $p$-subgroup and ...
3
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1answer
57 views

Prove that if $(12)(34),(12345)\in G$, it is not solvable

Let $G$ be a group acting on $X=\{a,b,c,d,e\}$. Given that there exist two elements $g,h\in G$ so that $$ g(a)=b,\ g(b)=c, \ g(c)=d, \ g(d)=e, \\ h(a)=b, \ h(b)=a, \ h(c)=d, \ h(d) =c. $$ Show ...
3
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1answer
30 views

Finite groups all of whose subgroups are CLT

A $CLT$-group is a finite group with the property that for every divisor of the order of the group, there is a subgroup of that order (converse lagrange theorem). I know that: There is a $CLT$-group ...
2
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1answer
58 views

Why Fitting subgroup of finite solvable group is self-centralizing

I read the proof of the proposition that the Fitting subgroup of a finite solvable group is self-centralizing. But I do not understand why $B$ in the proof is characteristic. The proof says that $F (...
3
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1answer
57 views

$G$ is a group with a normal subgroup $K$ such that $G/K$ is soluble, and $H$ is a nonabelian simple subgroup of $G$, then $H \leq K$

I'm trying to see why the following theorem is true: If $G$ is a group with a normal subgroup $K$ such that $G/K$ is solvable, and $H$ is a nonabelian simple subgroup of $G$, then $H \leq K$. My ...
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0answers
29 views

Maximal normal series length for finite solvable groups

Aluffi IV.3.2 goes as follows: Let $G$ be a finite cyclic group. Compute $\ell(G)$ in terms of $|G|$. Generalize to finite solvable groups. Here $\ell(G)$ is the maximal length of a normal series ...
4
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2answers
130 views

Definition of polycyclic groups

I'm trying to understand the definition of polycyclic groups. A solvable group $G$ has two equivalent definitions: $G$ has a subnormal series like $$G = H_n \rhd H_{n-1} \rhd \cdots \rhd H_0 = 1$$ ...
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0answers
21 views

A certain quotient of a (not necessarily solvable) finite group is solvable

Let $G$ be a finite group, which is not necessarily solvable. I am trying to prove the following statement: There exists a smallest normal subgroup $N(G) \lhd G$ such that $G/N(G)$ is solvable I ...
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1answer
31 views

Invariant Sylow subgroups

Today, I'm reading lemma 2.2.c of an article by Antonio Beltran. Lemma 2.2. Suppose that A is a finite group acting coprimely on a finite group G, and let $C = C_G(A)$. Then, for every prime p, c) if ...
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2answers
203 views

Solvable Group of Polynomial of Degree 4

I am studying the insolvability of polynomial of degree $>4$, i.e., indeterminates more than $4$ in the the article Galois Theory for Beginners by John Stillwell. The complete proof is given below. ...
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Construct normal series from derived series through generating sets

This is from section 2 of the paper "Quantum algorithms for solvable groups" by J. Watrous, where $G$ is a solvable group, and $G = G^{(0)} \rhd G^{(1)} \rhd \cdots \rhd G^{(n)}$ is the derived series ...
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1answer
35 views

Find an extension $F/L$ such that $F/\mathbb{Q}$ is radical

Let $f(x)= x^3 - 7x +7 \in \Bbb Q[x]$ and let $L/\Bbb Q$ be the splitting field of $f$. I have shown that the Galois group of $L/\Bbb Q$ is $C_3 $ and so as this is solvable, $f$ is solvable by ...
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1answer
25 views

Decreasing sequence of normal subgroups of a group $G$

Let $n\in\mathbb{N}$ and let $G$ be a group. If there exists a sequence of normal subgroups of $G$ $$G=G^{0}\supset G^1\supset\ldots\supset G^n=\{e\}$$ such that the groups $G^k/G^{k+1}$ are ...
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0answers
22 views

If g is a solvable Lie algebra then its form of Killing is zero? [duplicate]

I know this is not true but I would like to know a counterexample. This is contrary implication of Cartan's criterion. I would also like someone to tell me what other conditions the soluble lie ...
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1answer
45 views

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

This is a article which Gabriel Navarro wrote. I'm reading lemma 2.1. I see that "$|C ∩ P : C ∩ Q|≤|P : Q|$, and therefore $|C ∩ P : C ∩ Q|$ divides $|P : Q|$. We deduce that $|C|q/|C ∩ H|q$ divides ...
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1answer
92 views

Is there a criterion for which $BS(m,n)$ are solvable (and non-solvable)? If not, are there classes of such groups where this is known?

Let $BS(m,n) = \langle a,t\mid ta^mt^{-1} = a^n \rangle$ be a Baumslag-Solitar group, with $m,n \in \mathbb{Z}.$ Is there a criterion for which $BS(m,n)$ are solvable (and non-solvable)? If not, ...
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2answers
163 views

How can I solve, for example, $n^5-625n+1632=0$ for $n$ if it is solvable?

I understand that some quintics in Bring-Jerrard form are solvable but first one must identify a solvable group for it or any quintic. I don't know how to identify a group for this equation or how to ...
3
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0answers
90 views

Hall's theorem and generalized Sylow theorem.

Hall's theorem states that: Let $G$ be a finite group. The following statements are equivalent: $G$ is solvable. $G$ is $\pi$-separable for every set of primes $\pi$. $G$ contains a $\pi$-Hall ...
6
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1answer
101 views

Can any solvable finite group be obtained from abelian groups and combinations of taking subgroups, quotients, and semidirect products?

It is clear that finite solvable groups are closed under those operations, so at most the solvable groups can be produced this way. Not all solvable groups can be written as semidirect products of ...
2
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0answers
65 views

Why is the free group on two generators not a subgroup of $G$?

Let $G$ be generated by the elements $g$ and $\{e_i\}$ for $i\in\mathbb{Z}$, having the relation $ge_ig^{-1}=e_{i+1}$. It seems to me like there is a subgroup $\langle e_i,e_{i+1}\rangle$ for example,...
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0answers
65 views

How can I solve a Bring-Jerrard quintic equation? [duplicate]

I have an equation $$n^5-m^4n+\frac{P}{2m}=0\text{ where 60 divides $P$ and where $m\in\mathbb{N}\land m>1$}$$ I want to solve for $n$, as in $n=f(P,m)$. I will know the value of $P$ and a range ...
6
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4answers
575 views

How do I solve the quintic $n^5-m^4n+\frac{P}{2m}=0$ for $n$?

I want to solve the following equation for $n$ in terms of $P$ and $m$. $$n^5-m^4n+\frac{P}{2m}=0$$ I've bought and read many books, including "Beyond The Quartic Equation" but I've either missed ...
2
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1answer
77 views

Problem from Galois' personal works

Let $R =\mathbb Z /p\mathbb Z$, with $p $ prime, and consider the set $A $ of the permutations of $R $ of the form $$\rho_{a,b}:x \mapsto ax+b ,$$ with $a,b \in R$ and $a \ne 0$. Call $T $ the subset ...
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1answer
59 views

Slight reordering of composition factors in a finite solvable group of order 4 mod 8

Given a solvable finite group G of order 4 mod 8, I would like to collect the 2-torsion in adjacent positions, if possible. The Sylow 2 subgroup has order 4, and so is either cyclic or the Klein 4-...
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2answers
46 views

Show that a polynomial is solvable (soluble)

So I've recently encounter the problem: Show that the Galois group of $x^{10}-2$ over Q is solvable I've been trying tp show that the splitting field of this pol is contained in a radical ...
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1answer
58 views

On solvable equations and a good book about Galois Theory

Well, as the title says I would like to know a good and direct book about Galois Theory (and also that apllies the theory for the study of polynomials). It's been kinda impossible for me to find such ...
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1answer
36 views

The composition length equals the sum of the exponents on all prime divisors in a prime factorization.

I read that "The composition length equals the sum of the exponents on all prime divisors in a prime factorization." in this link: https://groupprops.subwiki.org/w/index.php?title=...
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1answer
30 views

Normal Subgroups of Solvable Groups with elementary abelian Sylow 2 subgroup

I am trying to understand implications of the composition series for groups with elementary abelian Sylow 2 subgroups, to see how much of the odd part of the group can be separated. Suppose G has ...
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1answer
33 views

Complement for a minimal normal subgroup in a solvable group

Let $G$ be a solvable group, $N$ a minimal normal subgroup of $G$ and suppose that exist $H$ a proper subgroup of $G$ such that $G=NH$. I want to prove that there is a complement of $N$ in $G$. Right ...
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1answer
40 views

Feit-Thompson implies nonexistence of simple groups of odd composite order?

In Dummit and Foote, Section 6.2 (Applications in Groups of Medium Order), it states that the Feit-Thompson Theorem asserts that there are no simple groups of odd composite order. The Feit-Thompson ...
3
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2answers
45 views

Normalizer of a solvable maximal subgroup

Let $G$ be a finite group and let $H$ be a solvable maximal subgroup of $G$, meaning that the only solvable subgroup of $G$ containing $H$ is $H$. I am trying to prove that $H=N_G(H)$. Since $H$ is ...
1
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1answer
46 views

If G is a solvable group whose partially ordered set of subgroups satisfies both the ascending and descending chain conditions, then G is finite.

Suppose $G$ is a solvable group whose partially ordered set of subgroups satisfies both the ascending and descending chain conditions. GOAL: Show that $G$ is finite. Since $G$ is solvable, there ...
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0answers
46 views

Commutator subgroup is strictly smaller if the center is nontrivial

I am trying to prove that every $p$-group is solvable, i.e., $G$ is solvable if $|G|=p^k$ for a prime $p$ and integer $k$. I want to show that if the commutator subgroup $G^{(1)}$ is strictly smaller ...
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1answer
114 views

Different Definitions for Solvable Groups

Dummit and Footes definition of solvable groups is $G$ has a chain of subgroups: $1=N_0\trianglelefteq N_1 \trianglelefteq N_2 \trianglelefteq \dots \trianglelefteq N_t = G$ such that each $N_i$ ...
5
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2answers
110 views

Is the join of two solvable subgroups solvable?

It is an exercise in Rotman's "introduction to the theory of groups" to show that if $S \trianglelefteq G$ and $T \leq G$ are solvable, then so is $ST$. I know $ST = S \vee T$ in the lattice of ...
4
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1answer
118 views

If maximal subgroups of solvable group have equal cores, then they are conjugates

I have been asked to Show that if $U$ and $V$ are maximal subgroups of a soluble group $G$, then the following conditions are equivalent: (a) - $U$ and $V$ are conjugates (b) - $U_{G} = V_{G}$, i.e., ...
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0answers
20 views

Group Theory: Solvable groups and commutators [duplicate]

Question : Show that a group G is solvable if and only if there exist sub-groups $G^{(0)}$, $G^{(1)}$, ..., $G^{(m)}$, with m $\ge$ 1, such that : $G^{(m)}$ = {$e_G$} and the $G^{(i)}$ are defined ...
0
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0answers
37 views

Assumption in Proof that Every $p$-Group is Solvable

I found this proof here My question is with fourth bullet point. It says, Note that $G_1 = Z(G)$ is normal in $G$. . . .So the quotient group $G/G_1$ is defined and furthermore $|G/G_1| = |G|\; /...
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0answers
27 views

Sylow, Suzuki group and solvability of finite groups

I read this theorem in the article which Antonio Beltran, I don't understand why prove this theorem: "we can easily prove (by induction on the order) that if a group G satisfies $ν_3(G)=1$ and has no ...

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