1
vote
1answer
27 views

Characterization of solvable groups in terms of subgroups of certain orders?

In this question, the OP mentions the following result: a finite group $G$ is solvable if and only if $$\text{for all $n$ dividing $|G|$ such that $\gcd(\frac{|G|}{n},n)=1$, $G$ has a subroup order ...
1
vote
1answer
42 views

Standard reference for Kaluzhnin's theorem

Kaluzhnin's theorem says that if $G$ is a group and $H \leq \operatorname{Aut}G$ acts trivially on each step of the normal series $1 = G_0 \leq G_1 \leq \ldots \leq G_n = G$, then $H$ has nilpotency ...
24
votes
1answer
341 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 ...