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

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4
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1answer
27 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
61 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
24 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
41 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
27 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
24 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
16 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
43 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
26 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
32 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
24 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
106 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
88 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
11 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
48 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 ...
0
votes
1answer
34 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
45 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
35 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
21 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
43 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
24 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
95 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
93 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
40 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
71 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
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0answers
31 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 ...
1
vote
2answers
90 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
89 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
56 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
30 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
139 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
39 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) ...
0
votes
1answer
46 views

Carter Subgroups and Projectors

R. Carter prooved that in finite soluble groups $G$ Carter subgroups $C$ exist and that they are conjugated. Furthermore they are exactly the nilpotent projectors: For every normal subgroup $N$ of $G$ ...
0
votes
2answers
127 views

group of order 135 solvable?

I need to prove that a group G of order 135 is solvable. $|G| = 135 = 5\cdot 3^3$ i found that the Sylow-subgroups are unique, so they are both normal. Let H be the Sylow-5-subgroup and F be the ...
0
votes
0answers
42 views

Locally graded groups

I have to prove that a solvable group is locally graded and I have thought to proceed by induction on the derived length of G. However, I can not prove the inductive basis. If der(G) = 1, G is ...
7
votes
3answers
125 views

How can I tell if $x^5 - (x^4 + x^3 + x^2 + x^1 + 1)$ is/is not part of the solvable group of polynomials?

I have developed an interest in generalisations of the fibonacci sequence, from tribonacci sequence up to what I'll coin the 'infinibonacci' sequence. I'm aware that these nth-bonacci sequences ...
1
vote
1answer
100 views

Solvable groups

I am practicing the concepts of solvable groups. I need help making sure my proof is correct and understanding an example from my lecture notes. Show that every group of order $500$ is solvable. ...
4
votes
2answers
105 views

Subgroups of smallest possible index in a solvable group

The following question appears in Isaacs' Finite Group Theory: 3B.15) (Berkovich) Let $G$ be solvable, and let $H<G$ be a proper subgroup having the smallest possible index in $G$. Show that ...
1
vote
1answer
115 views

Ideals of solvable lie algebra

Let us say that S is a Lie algebra of dimension $n$, which is also solvable. Is it true that S contains an ideal of each dimension $d$ for $0 \leq d \leq n$? If so, how? Thanks for all the help.
2
votes
3answers
94 views

Some solvable Lie algebra but not nilpotent

Can someone provide two concrete examples the Lie algebra which is solvable, but not nilpotent? -- And further explain the subtle differences between the solvable Lie algebra and the ...
1
vote
1answer
87 views

Prove $G$ is cyclic if for every $n\in \Bbb{N}$, there are at most $n$ solutions to the equation $x^n=e$. [duplicate]

Let $G$ be a finite group. Prove G is cyclic if for every $n\in \Bbb{N}$, there are at most $n$ solutions to the equation $x^n=e$. I am stuck with this :( Would appreciate your help.
2
votes
1answer
80 views

What is set builder of $\langle H, K \rangle$?

I am looking left and right for a lemma to solve a problem on solvable group here, and I think I have found one under commutator group: For any two subgroups $H, K$ of $G$, the $[H, K]$ is a ...
1
vote
2answers
161 views

Reconciling Different Definitions of Solvable Group

Looks like my class note defines solvable group differently from others: A finite group $G$ is called solvable if $H' \neq H$ for each subgroup $H$ of $G$ different from $\{1\}$, where $H'$ is ...