# Questions tagged [solvable-groups]

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

345 questions
Filter by
Sorted by
Tagged with
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 ...
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 ...
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 ...
44 views

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/...
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 ...
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 $n$
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 $H(F)/Z(H(F))$ ...
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 ...
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 $G$ ...
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 ...
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 ...
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 ...
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 $F (... 1answer 57 views ###$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$. My ... 0answers 29 views ### Maximal normal series length for finite solvable groups Aluffi IV.3.2 goes as follows: Let$G$be a finite cyclic group. Compute$\ell(G)$in terms of$|G|$. Generalize to finite solvable groups. Here$\ell(G)$is the maximal length of a normal series ... 2answers 130 views ### Definition of polycyclic groups I'm trying to understand the definition of polycyclic groups. A solvable group$G$has two equivalent definitions:$G$has a subnormal series like $$G = H_n \rhd H_{n-1} \rhd \cdots \rhd H_0 = 1$$ ... 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 ... 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 ... 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. ... 0answers 12 views ### 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 ... 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$is solvable by ... 1answer 25 views ### Decreasing sequence of normal subgroups of a group$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 ... 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 ... 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 ... 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, ... 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 ... 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 ... 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 ... 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,... 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 ... 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 ... 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 $the subset ... 1answer 59 views ### Slight reordering of composition factors in a finite solvable group of order 4 mod 8 Given a solvable finite group G of order 4 mod 8, I would like to collect the 2-torsion in adjacent positions, if possible. The Sylow 2 subgroup has order 4, and so is either cyclic or the Klein 4-... 2answers 46 views ### Show that a polynomial is solvable (soluble) So I've recently encounter the problem: Show that the Galois group of$x^{10}-2$over Q is solvable I've been trying tp show that the splitting field of this pol is contained in a radical ... 1answer 58 views ### On solvable equations and a good book about Galois Theory Well, as the title says I would like to know a good and direct book about Galois Theory (and also that apllies the theory for the study of polynomials). It's been kinda impossible for me to find such ... 1answer 36 views ### The composition length equals the sum of the exponents on all prime divisors in a prime factorization. I read that "The composition length equals the sum of the exponents on all prime divisors in a prime factorization." in this link: https://groupprops.subwiki.org/w/index.php?title=... 1answer 30 views ### Normal Subgroups of Solvable Groups with elementary abelian Sylow 2 subgroup I am trying to understand implications of the composition series for groups with elementary abelian Sylow 2 subgroups, to see how much of the odd part of the group can be separated. Suppose G has ... 1answer 33 views ### Complement for a minimal normal subgroup in a solvable group Let$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$. Right ... 1answer 40 views ### Feit-Thompson implies nonexistence of simple groups of odd composite order? In Dummit and Foote, Section 6.2 (Applications in Groups of Medium Order), it states that the Feit-Thompson Theorem asserts that there are no simple groups of odd composite order. The Feit-Thompson ... 2answers 45 views ### Normalizer of a solvable maximal subgroup Let$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$is ... 1answer 46 views ### If G is a solvable group whose partially ordered set of subgroups satisfies both the ascending and descending chain conditions, then G is finite. Suppose$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$is solvable, there ... 0answers 46 views ### Commutator subgroup is strictly smaller if the center is nontrivial I am trying to prove that every$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)}$is strictly smaller ... 1answer 114 views ### Different Definitions for Solvable Groups Dummit and Footes definition of solvable groups is$G$has a chain of subgroups:$1=N_0\trianglelefteq N_1 \trianglelefteq N_2 \trianglelefteq \dots \trianglelefteq N_t = G$such that each$N_i$... 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 ... 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., ... 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)}$are defined ... 0answers 37 views ### Assumption in Proof that Every$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|\; /...
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 ...