# Tagged Questions

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

569 views

### 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 ...
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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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### Group of order $8p$ is solvable, for any prime $p$

Consider the following question: Show that a group $G$ of order $8p$ is solvable, for any prime $p$. I am kind of stuck, but here are my first attempts: I chose the series of subroups ...
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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?
### Class of $p$-supersolvable group is saturated formation.
I want to show that the class of $p$-supersolvable groups is a saturated formation. I only have the definition of $p$-supersolvable. What do I do? A group is $p$-supersolvable iff every chief factor ...