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

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2
votes
1answer
36 views

Is there a proof that this polynomial solvable by radicals?

If $f(x)$ is the minimal polynomial of $t$, a constructible number, over $\mathbb{Q}$, then is $f(x)$ solvable by radicals? It seems to be true, at least with the examples I came up with. Can it be ...
1
vote
1answer
28 views

$G$ is solvable if and only if it has a normal series with each factor of order a power of a prime

Let $G$ be a group with a composition series. Then $G$ is solvable if and only if $G$ has a normal series (which starts at $1$) and each factor group has order some power of a prime. I think in ...
-1
votes
1answer
40 views

Let $F\subset E$. If $f(x)$ is solvable over $F$, prove that $f(x)$ is solvable over $E$. [closed]

I'm trying to show that if $f(x)$ is solvable over $F$, then $f(x)$ is also solvable over $E$. It's intuitively true, but I don't know how to show it rigorously.
1
vote
0answers
17 views

If a group $G$ is solvable, is the subnormal series (the chain of the normal subgroups) unique?

If a group $G$ is solvable, is the subnormal series (the chain of the normal subgroups) unique? Or what type of solvable groups have unique subnormal series?
1
vote
2answers
61 views

Why is the group of unit upper triangular matrices solvable?

Let $GL_n(k)$ be the n by n general linear group over $k$, $B_n(k)$ be the subgroup of $GL_n(k)$ consisting of all upper triangular matrices, and $U_n(k)$ be the subgroup of $B_n(k)$ whose diagonal ...
0
votes
0answers
32 views

Can the number of non-solvable groups of a given order be easily determined?

It can be extremely difficult to find the number of groups of a given order. But if we only want to find the number of non-solvable groups of a given order, is there an easy algorithm doing the job ...
1
vote
1answer
46 views

Is there a non-solvable number NOT divisible by $3\ $?

Here https://oeis.org/A056866 it is claimed that every non-solvable number is divisible by $4$ and either $3$ or $5$. However, I did not find a number in the list not divisible by $3$. So, my ...
2
votes
1answer
109 views

$G$ contains a normal $p$-subgroup

Let $G$ be a non-abelian finite group with center $Z>1$. I want to show that if $G/Z$ is solvable then $G$ contains a normal $p$-subgroup for some prime $p$ with $p\mid |G:Z|$. $$$$ Since ...
1
vote
3answers
48 views

Solving $X x^t+ Y y^t=1$ for a specific case with constraints

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 ...
4
votes
1answer
65 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 ...
0
votes
1answer
23 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 ...
4
votes
1answer
33 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 ...
3
votes
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 $n_2 \in \{1,3\}$ ...
1
vote
0answers
30 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 ...
2
votes
1answer
46 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 ...
0
votes
1answer
42 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 ...
0
votes
1answer
30 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 ...
0
votes
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 ...
0
votes
1answer
45 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.
1
vote
1answer
34 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 ...
1
vote
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 = ...
1
vote
0answers
25 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 $G_{i+1}/G_i$ ...
3
votes
2answers
114 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?
3
votes
0answers
93 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 ...
1
vote
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 ...
0
votes
2answers
56 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 ...
1
vote
1answer
44 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 ...
0
votes
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, ...
0
votes
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 ...
1
vote
3answers
51 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 $G$, ...
0
votes
1answer
36 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 $, ...
0
votes
2answers
39 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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
25 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 ...
2
votes
2answers
149 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 ...
3
votes
1answer
122 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 ...
0
votes
0answers
41 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 ...
0
votes
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 ...
1
vote
1answer
77 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 $, ...
0
votes
1answer
35 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 $d$?
1
vote
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 ...
2
votes
2answers
110 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 ...
0
votes
1answer
91 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 ...
-1
votes
1answer
64 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 ...
2
votes
2answers
145 views

subtracting inequalities if difference is positive

$$ x \geq y $$ $$ a \geq b $$ $$x+a \geq y+b $$ 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 ...
0
votes
1answer
35 views

Derived length of direct product of 2 soluble groups

I'm struggling with an assignment question on the topic of soluble groups. The question is to prove that if $G = H \times K$ is a soluble group, and $H$ and $K$ have derived lengths $n$ and $m$ ...
2
votes
2answers
209 views

Solvable equivalent to nilpotency of first derived Lie algebra?

The Wikipedia "Solvable Lie Algebra" page lists the following property as a notion equivalent to solvability: $\mathfrak{g}$ is solvable iff the first derived algebra $[\mathfrak{g},\,\mathfrak{g}]$ ...
1
vote
2answers
41 views

Show that if $A, B$ are groups, then $A \times B$ is solvable if and only if both $A$ and $B$ are solvable? [closed]

Show that if $A, B$ are groups, then $A \times B$ is solvable if and only if both $A$ and $B$ are solvable? Why is obvious that if $A$ and $B$ are solvable then $A \times B$ is solvable?
0
votes
1answer
49 views

Consequences of $G$ having exactly $3$ irreducible characters

Suppose a finite group $G$ has exactly irreducible characters $\chi_1 = \mathbb{I}, \chi_2,\chi_3$. i) Show that G is soluble and deduce it has a non-trivial one-dimensional character $\chi_2$. ii) ...