# Tagged Questions

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

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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,\...
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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 $\ge 5$ is not solvable using radicals, unlike the polynomial equation of second degree which is solvable by radicals (because of the ...
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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 ...
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### 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 $[\mathfrak{g},\,\mathfrak{g}]$ ...
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### 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 ...
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### Finite Group soluble $\iff G^{(k)}=\{e\}$ for some $k$

I am trying to understand the proof of the following proposition: A finite group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$, where $G^{(0)}=G$, $G^{(i+1)}=[G^i G^i]$ is the derived series. ...
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### Every minimal normal subgroup of a finite solvable group is elementary abelian

Show that every minimal normal subgroup of a finite solvable group is an elementary abelian $p$-group for some prime $p$. I'm stuck on this one, any idea is appreciated.
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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 ...
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### Finite groups with all maximal subgroups of prime power index are solvable?

It is well known that in finite solvable groups, any maximal subgroup has prime power index. Question: If $G$ is a finite group in which any maximal subgroup has prime power index, is $G$ ...
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### 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 ...
My question is straightforward to pose: given a polynomial $f$ over a subfield of $\mathbb{C}$, are there conditions which guarantee the existence of a closed formula for the roots of $f$ in terms of ...