# Tagged Questions

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

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### 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?
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### Derived algebras and solvable Lie algebras

The idea of a Solvable Lie algebra hinges on the definition of the sequence: $$g \ge [g,g] \ge [[g,g],[g,g]] \ge [ [[g,g],[g,g]] , [[g,g],[g,g]] ] \ge \ldots$$ and its limiting group. If an ...
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Is there an analytical solution to the equation below? \begin{align*} A\frac{\alpha}{k}e^{(k-\alpha)t}+B\frac{\beta}{k}e^{(k-\beta)t}=1 \end{align*} where $\alpha$ and $\beta$ are roots of $$\... 1answer 68 views ### Are these groups solvable? I am thinking of Baumslag-Solitar groups of type BS(1,m)=\langle a,b \mid bab^{-1} = a^m\rangle as a prototype. We can think of them as follows: Start with an infinite cyclic group \langle a\... 1answer 25 views ### A finite p-group has a supersolvable series. I'm being asked to show: Show a finite p-group G has a supersolvable series, i.e. a normal series$$G=G_0\ge G_1\ge\cdots\ge G_m=1$$such that each factor group is cyclic and each G_i is ... 1answer 35 views ### Group theory commutator and solvable groups let G be a group such that it contains 2 members a, b \in G that statisfy: a = p^{-1} b p where p \in G a = q^{-1} [a,b]q  where q \in G a,b,[a,b]\neq e where [a,b] is the commutator ... 1answer 68 views ### prove that GL_2(\Bbb Z_3) is solvable I need to prove that GL_2(\Bbb Z_3) is solvable What I tried: I know that GL_2(\Bbb Z_3) has (3^2-1)(3^2-3) = 48 = 3 * 2^4 elements. I know that n_3 \in \{1,4,16\} and n_2 \in \{1,3\} ... 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 ... 1answer 47 views ### Is this group simple? I have this group presentation G = \langle a,b | ba = a^{-1}b\rangle. I'm wondering if this group is solvable or not. I tried to show that this group is simple, because if it is, then it can not be ... 1answer 53 views ### Dihedral groups are solvable [duplicate] I'm trying to prove the dihedral groups are solvable for any Dn. I use the normal subgroup of all rotations, since the quotient of Dn/{rotations} is isomorphic to Z2 so it's abelian as well, so we get ... 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 ... 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 ... 1answer 49 views ### Why simply connected solvable analytic groups have no nontrivial compact subgroups? Why do simply connected solvable analytic groups have no nontrivial compact subgroups? I'll appreciate any help on this question. 1answer 42 views ### Self normalizing maximal subgroup of a non solvable group Let G be a finite non-solvable group and H its maximal subgroup. Prove that if H is solvable then H=N_G(H) I think I found different ways to prove it but I don't know how to begin: -if N_G(H)=... 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\\... 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 ... 2answers 118 views ### A proof that BS(1,2) is not polycyclic I am looking for examples of finitely generated solvable groups that are not polycyclic. In Wikipedia Baumslag-Solitar group BS(1,2) is an example. But how to prove this fact? 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 ... 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-... 2answers 59 views ### Finite group is solvable iff it has a normal series with p-primary abelian quotient groups Let G be a finite group. It is stated that group G is sovable \iff there exists a normal series$$\{e\}=H_s \lhd H_{s-1} \lhd \cdots \lhd H_{1} \lhd H_0 = G$$such that each quotient ... 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) ... 1answer 45 views ### Why is \Bbb{Z_2} / \{e\} = \Bbb{Z_2}? Let the group \Bbb{Z_2} = \{e, a\}. We are given the quotient group \Bbb{Z_2} / \{e\}. So this gives us a set of left cosets: \Bbb{Z_2} / \{e\} = \{e\{e\}, a\{e\}\} = \{\{e\}, \{a\}\} \neq \{e, a\... 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 ... 3answers 52 views ### If G is a group such that any two commutators commute, G is solvable I need to prove that if G is a group such that any two commutators of elements of G commute, then G is solvable. This is the idea that I had: The subgroup of all commutators of G, G^{\prime}... 1answer 40 views ### show if  G  is supersoluble then  G  has a chief factor such that every chief factor of order  p . We have this theorem: Let  G  be a non-trivial finite supersoluble group. Then  G  has a normal series  1 = G_{0} <G_{1} < \cdots < G_{s} = G , such that for every  1 \leq i \leq s , ... 2answers 40 views ### Question about upper central series of a group I am trying to figure out nilpotent groups, and it says that a group G is nilpotent if there is a non-negative integers c such that Z^c(G) = G. Now an upper central series is something like Z^... 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 -... 1answer 46 views ### H, N subgroups of S_{5} This is a homework problem but I have stuck at some other chapter earlier so I am now completely lost what I am supposed to do. Clues and hints and suggestions of theorems that I should look up would ... 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.... 2answers 174 views ### Group of order 48 must have a normal subgroup of order 8 or 16 Prove a group of order 48 must have a normal subgroup of order 8 or 16. Solution: The number of Sylow 2-subgroups is 1 or 3. In the first case, there is a normal subgroup of order 16... 1answer 127 views ### G solvable \implies composition factors of G are of prime order. I'm trying to prove the equivalency of the following definitions for a finite group, G: (i) G is solvable, i.e. there exists a chain of subgroups 1 = G_0 \trianglelefteq G_1 \trianglelefteq \... 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 ... 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 ... 1answer 78 views ### Assume that  G = MC , for some cyclic subgroup  C . Is  M \cap C  a normal subgroup of  G ? Let  G  is a solvable finite group and  M  be a maximal subgroup of  G , and assume that  G = MC , for some cyclic subgroup  C . If  M_{G} = 1  that  M_{G}  is core of  M  in  G , is... 1answer 37 views ### existence of a subgroup of a solvable group G of order d for any divisor d of |G|, with d<|G|^(1/2) Let G be a solvable group of order n and d<\sqrt{n} be any divisor of n. Is there any subgroup of G of order d? 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)},\... 2answers 119 views ### Quotients of Solvable Groups are Solvable I just proved that subgroups of solvable groups are solvable. So given that G is solvable there is 1=G_0 \unrhd G_1 \unrhd \cdots \unrhd G_s=G where G_{i+1}/G_i is abelian and for N a normal ... 1answer 92 views ### Solvability of transitive group Let G be a transitive subgroup of S_p where p is an odd prime number. Now consider the following assumptions - (i) G is solvable. (ii) If \sigma \in G and there exist h\ne j such ... 1answer 67 views ### Soluble(solvable) and nilpotent groups Defn 1.1. Let \gamma _{0}(G)=G, and \gamma _{c}(G)=[\gamma _{c-1}(G),G] for c\geq 1. The lower central series of G is a chain of subgroups of G:$$G=\gamma_0(G) \geq \gamma_1(G) \geq \cdots \...
$$x \geq y$$ $$a \geq b$$ $$x+a \geq y+b$$ is valid but $$x-a \geq y-b$$ is not valid Can we say the latter is valid if $x-a \geq 0$ ? Is it a proof or am I wrong? Are there counter examples?...