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Questions tagged [solvable-groups]

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

2
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1answer
27 views

solvable groups have at least two irreducible representations of dim 1

Let G be a finite group and $n$ the number of irreducible characters of dimension $1$ of G. Prove that $n>1$. The hint I have is not convincing or somewhat unclear: "$G^{ab}:=G/[G, G]$ is non ...
0
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0answers
94 views

On the solvability group

Let $G$ and $H$ be two groups such that $|G|=|H|$; for every natural number $n$ the number of elements of order $n$ in $G$ and $H$ are equal; $H$ is a solvable group. Is $G$ solvable? or Is there ...
5
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1answer
65 views

Proof of $G$ is solvable implies $G/N$ is solvable.

I want to show that if $N$ is normal in $G$ then $G$ is solvable implies $G/N$ is solvable. Now, $G$ is solvable implies there exists a chain $\{e\}=G_0 \trianglelefteq G_1 \trianglelefteq G_2 \...
1
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0answers
30 views

Definition of solvable group in Fraleigh book

I have been using Fraleigh's book, A first course in abstract algebra There is one thing I want you to ask whether the definition of solvable group in this book is right or not. In my book, ...
1
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1answer
25 views

Set of Rotations Cyclic?

For the dihedral group $D_{n}$ of order $2n$, is the group $R$ formed by its $n$ rotations cyclic in general? Or is the factor group $D_{n}/R$ cyclic? I am trying to show the series $D_{n}>R>(1)$...
0
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0answers
21 views

Derived series of the Dihedral group [duplicate]

I'm working on derived subgroups because I'm studying for an exam and I want to show that in the case of the dihedral group $D_{2n}=\langle\sigma ,\tau|\sigma^n=\tau^2,\sigma^{\tau}=\sigma^{-1}\rangle$...
3
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1answer
28 views

Two questions regarding this proof from my notes which shows that if $H$ is normal in $G$ and $G$ is solvable then $\tfrac{G}{H}$ is solvable

Consider the following theorem : Let $G$ be a group. If $H$ is normal in $G$, and $G$ is solvable then $\tfrac{G}{H}$ is solvable. According to my lecture notes the proof proceeds as follows: If $H$...
21
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1answer
206 views

On automorphisms of groups which extend as automorphisms to every larger group

For a group $G$, let $\operatorname{Aut}(G)$ denote the group of all automorphisms of $G$ and $\operatorname{Inn}(G)$ denote the subgroup of all autmorphisms which is of the form $f_h(g)=hgh^{-1}, \...
4
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1answer
51 views

Abelian normal subgroups of A-groups

Let $G$ be a finite solvable group, where every Sylow subgroup is abelian. I want to show that if $A\lhd G$ is an abelian normal subgroup, then $$ A=(A\cap Z(G))(A\cap G')$$ This is easy if $A$ is a ...
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1answer
92 views

Proof that a group of order 8888 is solvable [closed]

I need to prove that every group of order $8888$ is solvable by proving that $G^{(3)}=\{e\}$. Can you give some help?
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0answers
32 views

Relation between odd order theorem and the fact that every polynomial of odd order has at least one root

It is known that there is connection between solvability of a group and expressing the roots of a polynomial by radicals, which is something I will study in this semester; however, by the odd order ...
3
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1answer
46 views

If $G$ is linear over $\mathbb{Z}$ then $\operatorname{Aut}(G)$ is linear over $\mathbb{Z}$ as well?

If $G$ is linear over $\mathbb{Z}$ then $\operatorname{Aut}(G)$ is linear over $\mathbb{Z}$ as well? We have $G \cong H \leq GL(n, \mathbb{Z})$ for some $n$ (this is what I mean when I say that $G$ ...
-1
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1answer
50 views

Show that any finite simple non-abelian group is not solvable. [closed]

Aluffi's Algebra claims (implicitly) that any finite simple non-abelian group is not solvable. On the other hand, it defines a group to be not solvable if its derived series doesn't terminate with ...
1
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1answer
40 views

Flag in vector space $V$ and a group $G$ stabilising it. Why is $G$ solvable?

Let $V$ be a $k$-vector space of dimension $n$. Take a flag $$0 = V_0 \subsetneq V_1 ...\subsetneq V_{m-1} \subsetneq V_{m} = V , $$ and a subgroup $G \leq GL(V)$ which stabilizes this flag, so ...
4
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3answers
178 views

Odd order groups with order less than 1000 are solvable

When I read in Group Theory of Scott. It has a question, I think it's hard. I have tried to solve it, but I can't. Problem: "If G is a group whose order is odd and less than 1000, then G is solvable" ...
8
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0answers
96 views

Is there a categorification of “(virtually) solvable”?

If this question doesn't make sense or is otherwise poor quality, then I'm sorry. Motivation: As part of my research, I study virtually solvable (1) groups. These are goups that have a solvable ...
2
votes
1answer
74 views

The (un)decidability of the Tits Alternative for any given (suitably defined) set of groups.

Please forgive me if this question is ill-formed. I don't know much about decidability. Some Background: There are problems in combinatorial group theory that are undecidable, such as the word ...
0
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0answers
51 views

Splitting field of $x^p − x + t$ not solvable over F

Given $p$ is a prime, $k$ is an algebraically closed field of characteristic $p$. and $F = k(t)$, where $t$ is a variable, let $L$ be the splitting field of $x^p − x + t$ over $F$. Then it can be ...
0
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0answers
29 views

Galois group of Solvable extension is solvable. (Proof)

I gonna show that if $F\subset L$ is solvable and Galois the theorem above holds. Suppose $F\subset L$ is Galois and radical. Since L is radical over F, we have $F\subset F_1\subset F_2 ...\subset ...
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0answers
26 views

Proof of solvability of Galois group of Solvable extension.

I don't understand why we can assume that $F$ contains any primitive m roots of unity. Does it have anything to do with If $F \subset L$ is a radical extension, is $F(c_1,c_2,...,c_n)\subset L(c_1,...
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1answer
26 views

Question Regarding Solvability of finite group G.

I would like to clarify my proof(Sketch) for part A: We know that $G_{n-1}$ is finite and abelian(i.e. $G_{n-1}/\{e\}$ is abelian ) therefore it is solvable. Since, $G_{n-2}/G_{n-1}$ is finite and ...
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0answers
32 views

Supplement in finite groups

Let $G$ be a finite non-Abelian group. Is it true that `` there exists a proper solvable subgroup $A$ and a proper subgroup $B$ of $G$ such that $G=AB$?"
1
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1answer
33 views

A proof by induction about composition series

In above lemma and proof, I can't understand how do we consider $G$ fixed. Because, I think that if $G$ fixed then, for proof's argument, if $G$ has composition series of length $n$, then it has to ...
0
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0answers
77 views

What is the connection between simple groups, composition series, and solvable groups?

I am reading Dummit and Foote section 3.4 (Composition series and the Holder program) and I am having trouble understanding how these concepts are connected. A group $G$ is simple if the only normal ...
1
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3answers
72 views

$G$ is a finite solvable group. $M$ is a minimal nontrivial normal subgroup of $G$. Prove that $M$ is abelian

Hint is to construct a subgroup $N \subset M$ of prime index, and prove that commutators of M lie in N. There is an answer here but I'm not so familiar with commutators as I've only read about them ...
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0answers
350 views

Maximal subgroups that force solvability.

For which finite groups $M$ is it the case that every finite group $G$ with $M$ as a maximal subgroup solvable? If $M$ satisfies this condition then $M$ is solvable. Also, if $M$ is abelian then $M$ ...
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1answer
127 views

Solvable, non-nilpotent group with nilpotent commutator subgroup

What is the smallest example of a finite solvable, non-nilpotent group $G$, such that its derived subgroup $G'$ is nilpotent, but not abelian?
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2answers
67 views

How many distinct composition series does the group $D_{12}$ have?

How many distinct composition series does the group $D_{12}$ have? I know that $D_{12} \trianglerighteq \mathbb{Z}_6 \trianglerighteq \mathbb{Z}_3 \trianglerighteq \{e\}$ is a composition series ( ...
0
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1answer
105 views

Every group of order $2p^n$, when p is prime, is solvable

I have to prove that every group of order $2p^n$, p is prime, is solvable. When $n=1$, the group either is cyclic or dihedral, in any case the group is solvable. When $n=2$, if P is p-sylow subgroup,...
1
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2answers
73 views

$SL(6, \mathbb{C})$ is not solvable

Prove that $SL(6, \mathbb{C})$ is not solvable Generally I never proved that a group is not solvable... I thought about showing it is isomorphic to other groups which are not solvable (such as $S_6$)...
1
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1answer
75 views

Solvable quotients imply quotient of intersection is solvable

This is a question from an exam in an undergraduate group-theory course: Given a group $G$ and normal subgroups $N,K\trianglelefteq G$ such that $G/K,G/N$ are solvable, prove that $G/(K\cap N)$ is ...
2
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0answers
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Solvable, Supersolvable, and Nilpotent [duplicate]

I have learned the concepts for solvable, supersolvable, and nilpotent groups and their associated properties. In particular, we have $$\{\mbox{nilpotent groups}\}\subset\{\mbox{supersolvable groups}\}...
8
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1answer
171 views

Why are solvable groups important?

As we know from Galois theory, an irreducible polynomial is soluble in radicals if and only if its Galois group is solvable. However, solvable groups seem to have an importance in group theory far ...
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88 views

On the proof and equivalence statement of Feit-Thompson

This, I will explain, is not a duplicate of this I think because it has 2 parts and because I have some things to add on to it. There are 2 questions: (1) More in-depth explanation of the answer ...
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0answers
26 views

Can the triangularization of a linear group preserve a given good feature of an element?

Let $k$ be an algebraic-closed field and let $G$ be a soluble subgroup of $GL_n(k)$. By a result of Mal'cev (Bertram A.F. Wehrfritz - Infinite Linear Groups, Theorem 3.6), $G$ contains a normal ...
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0answers
57 views

A doubt on solvable groups and algebraic Number Theory

In Marcus book “Number Fields” I have this exercise: (page 124, number 25) Let $L$ be a normal extension of $K$ and suppose $K$ contains a prime which becomes a power of a prime in $L$. Prove that the ...
1
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1answer
74 views

If G is not abelian with order $p^3$ then $G'=Z(G)$

Question If $|G|=p^3$ and $G$ is not abelian show that $G'=Z(G)$ Attempt Since $|G|=p^3$ then $G$ is solvable and let $$1\leq G^{(n-1)}\leq...\leq G^{(n)}=G,(1)$$ be its derived series.We know ...
2
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1answer
72 views

If G is finite solvable then it has a normal subgroup of special form

Question Show that if $G$ is a finite solvable group then it contains a normal subgroup of the form $c_p\times...\times c_p$ with $p$ a prime. Attempt Since $G$ is solvable it contains a fully ...
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0answers
32 views

How to reduce a long list of rotations, translations and scalings down to four general transformations?

There are lots of points in 2D. All of them get their positions changed the same way by arbitrary numbers of random rotations, translations and scalings. - Rotations r(rx,ry,rad) rotate all points ...
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1answer
146 views

If a group $G$ has an abelian subgroup of index $2$, then $G$ must be solvable.

If a group $G$ has an abelian subgroup of index $2$, then $G$ must be solvable. Thoughts: I know that if $N$ is a subgroup of a group $G$ with $[G:N]=2$, then $N$ is normal. Context: Prof. Derek ...
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0answers
68 views

Group of order 24 is solvable. [duplicate]

I'm trying to show that any group of order 24 is solvable. I know $G$ is solvable if and only if both it's normal subgroup $N$ and $G/N$ is solvable. I have proved previously that a group of order 24 ...
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0answers
27 views

Suppose all the zeros of $f(x)$ are constructible. Show that $f(x)$ is solvable by radicals.

I know that a root $\alpha$ is constructible if [$Q(\alpha):Q$] = $2^n$ for some integer n. If $\Sigma$ is the splitting field of $f(x)$, $n \leq [\Sigma : Q] =Gal(f(x)) \leq n!$ so, in this case, $2^...
2
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1answer
44 views

Let $f(x)$ be irreducible in the rationals so that the order of $\operatorname{Gal}(f(x))$ is $255$. Show that $f(x)$ is solvable by radicals.

I know that $f(x)$ is solvable by radicals $\iff \def\Gal{\text{Gal}}\Gal(f(x))$ is solvable, so I began by trying to find what $\Gal(f(x))$ is. Let $G = \Gal(f(x)).$ Since $|G|= 255 = (3)(5)(17)$, I ...
2
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3answers
101 views

Prove that $GL_n(\mathbb{C}) $ is not solvable

How should I prove that $GL_n(\mathbb{C}) $, the group of invertible matrices over the complex numbers, is not solvable? I have no idea how to prove this by supposing that $GL_n(\mathbb{C}) $ is ...
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0answers
59 views

Compositum of solvable extensions

i have a question on solvable extensions. I am reading a paper where the authors work with the field $\mathbb{Q}^{solve}$, that is, the maximal solvable extension of $\mathbb{Q}$ in $\overline{\...
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1answer
80 views

Every group of order 300 is solvable?

I have a task which asks me to show, that every group of order 300 is solvable. But this should be false. When you take the direct product of the alternating group $A_5$ with $\mathbb{Z}/5\mathbb{Z}$,...
2
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1answer
71 views

Groups of Order 360

Show that every solvable group of order $360$ has a subgroup of order 90. (This is straightforward if you use the Philip Hall theorems. Is there an easy proof that doesn't use them?)
3
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1answer
187 views

Jordan -Holder for infinite series? composition series $\rightarrow$ chief series

The problem I am trying to prove is that a group which has a composition series will also have a chief series (we 're only talking about infinite groups of course). Instead of just posting the answer ...
2
votes
1answer
34 views

Is every group $G$ such that $\bigcap\limits_nG^{(n)}$ is trivial, solvable?

Suppose $G$ is a group, such that $\bigcap_{n=1}^\infty G^{(n)} = E$, where $G^{(n)}$ is the n-th element of the derived series of $G$, and $E$ is the trivial subgroup of $G$. Does $G$ always have to ...
2
votes
1answer
145 views

The context & motivation for the Tits alternative in combinatorial group theory

The Details: Definition 1: A class $\mathcal{G}$ of groups satisfies the Tits alternative if for any $G$ in $\mathcal{G}$ either $G$ has a free, non-abelian subgroup or $G$ has a solvable subgroup ...