Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [solvable-groups]

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

A group $G$ is called solvable if it has a subnormal series with abelian factors; that is, there are groups

$$\{1\} = G_0 \le G_1 \le G_2 \le \dots \le G_n = G$$

such that $G_i$ is normal in $G_{i + 1}$ and the factor group $G_{i + 1} / G_i$ is abelian for each $i$.

Solvable groups arise naturally in Galois theory, as a polynomial equation is solvable by radicals if and only if its Galois group is solvable.

Source: Solvable group.

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Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider $$G=\operatorname{Gal}(\mathbb{Q}(r_1,\ldots,r_n)/\mathbb{Q})$$ What is the probability, as…
Charles
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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 $M$ satisfies this condition. Futhermore, I believe…
Thomas Browning
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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 coefficients. But we also know that for degree > $n$,…
Assad Ebrahim
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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 $f_h(g)=hgh^{-1}, \forall g\in G$, where $h\in G$ . Now if $G_1$ is a…
user521337
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Why does "solvability" for groups suggest something about the solution of polynomials?

I watched a few videos about the solvability of a general quintic in radicals and I'm somewhat confused about a few concepts. My main confusion lies in the following definition; $\textbf{Def}:$ Solvable group A group $G$ is said to be solvable if…
Edward Evans
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For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?

I recently read the well known theorem that for a group $G$ and $H$ a normal subgroup of $G$, then $G$ is solvable if and only if $H$ and $G/H$ are solvable. In my book, only the fact that $G$ is solvable implies $H$ is solvable was proven. I was…
yunone
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Is the dihedral group $D_n$ nilpotent? solvable?

Is the dihedral group $D_n$ nilpotent? solvable? I'm trying to solve this problem but I've been trying to apply a couple of theorems but have been unsuccessful so far. Can anyone help me?
nomadicmathematician
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Is a polynomial equation of degree $\ge 5$ not solvable by any way?

By Galois Theory an arbitrary polynomial equation of degree greater or equal to 5 ($\ge 5$) is not solvable using radicals, unlike the polynomial equation of second degree which is solvable by radicals (because of the alternating group of order 5,…
Nikos M.
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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 beyond their implications for polynomial equations.…
Brian Bi
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If a finite group $G$ is solvable, is $[G,G]$ nilpotent?

If $G$ is a connected solvable Lie-group, then $[G,G]$ is nilpotent. The corresponding statement for Lie algebras follows from Lie's theorem, and it then follows from connected Lie groups by exponentiation. Is the statement also true for finite…
David E Speyer
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Solvable and nilpotent groups, normal series and intuition

I'm reading Hungerford's algebra and I'm on Nilpotent and solvable groups chapter. Hungerford starts with: Consider the following conditions on a finite group G: i) G is the direct product of its Sylow subgroups ii) If m divides |G|, then G has…
ante.ceperic
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When is $G \ast H$ solvable?

In a proof that the lamplighter group $\mathbb{Z}_2 \wr \mathbb{Z}$ is not finitely presented, I showed that $\mathbb{Z}_2 \ast \mathbb{Z}$ is not solvable. More precisely, one can prove that the commutator subgroup of $\mathbb{Z}_2 \ast \mathbb{Z}=…
Seirios
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Is there a solvable differential equation with a nonsolvable lie group of symmetries?

For a polynomial equation in one variable over $\mathbb{Q}$, it is well known that the equation is solvable by radicals if and only if the equation's Galois group (which is a finite group) is solvable. The 'only if' part is important - we need it to…
roymend
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What can be said about two groups with isomorphic derived factors?

The third isomorphism theorem states that we can relate an isomorphic relation between two normal subgroups of a group $G$. My question is can we infer anything about the two groups structures itself given that the factor/quotient groups in the…
Vaas
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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 elements are all $1$. To show $B_n(k)$ is…
JeongHobin
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