# Tagged Questions

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

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### Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
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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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### 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 ...
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### 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 ...
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### 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 $n_2 \in \{1,3\}$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 $G_{i+1}/G_i$ ...
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### 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?
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### 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 ...
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### 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-...
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### 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 ...
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### 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 $Z^c(G)$ ...
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### 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 $2$-...
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### $H, N$ 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 ...
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### 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....
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### 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 $16$...