Solvable Groups

Is every group of order 480 solvable?

Suppose \(f(x) \in \mathbb {Q}[x]\) is a polynomial with Galois group of order \(594 = 2 \cdot 3^3 \cdot 11\). Show that \(f\) is solvable in radicals.

Suppose \(G\) is a finite group with \(\operatorname {Aut}G\) solvable. Prove that \(G\) is solvable.

Let \(f(x)\) be a degree 5 polynomial in \(\mathbb {Q}[x]\) that is not solvable by radicals. Let \(L\) be its splitting field over \(\mathbb {Q}\).

1. Prove that there is at most one field \(K\) with \(\mathbb {Q}\subset K\subset L\) and \([K:\mathbb {Q}]=2\).
2. If \(\alpha ,\beta \) are irrational elements in \(L\) such that \(\alpha ^2\) and \(\beta ^2\) are rational, prove that \(\alpha \beta \) is rational.

Let \(G\) be the subgroup of \(\operatorname {GL}_2(\mathbb {R})\) consisting of matrices of the form \[ \begin {pmatrix} a & c\\ 0 & b \end {pmatrix}. \] Is \(G\) solvable?

Let \(U\) be the subgroup of \(\operatorname {GL}_2(\mathbb {Z}/p\mathbb {Z})\) consisting of upper triangular matrices. Is \(U\) solvable? Prove or disprove.

Let \(K\) be a field of characteristic zero. Let \(L\) be the splitting field of the polynomial \(X^n-a\) for some \(a\in K\) and some integer \(n\ge 1\). Show that the Galois group of \(L\) over \(K\) is solvable.

1. Let \(H\) be a maximal subgroup of the finite group \(G\) and let \(M\) be a minimal normal subgroup of \(G\). Assume that \(M\) is abelian but \(H\) does not contain \(M\). Prove that \(G=HM\), \(H\cap M=\{1\}\) (the identity subgroup), and \(|G:H|\) (the index of \(H\) in \(G\)) is a power of a prime.
2. Use the result in (a) to show that that if \(G\) is a finite solvable group and \(H\) is a maximal subgroup of \(G\), then \(|G:H|\) is a power of a prime.

Assume that \(A\) and \(B\) are solvable subgroups of the group \(G\) and \(G=AB\). Here \(AB=\{ab\mid a\in A,\ b\in B\}\).

1. Give an example showing that \(G\) does not have to be solvable.
2. Prove that if either \(A\) or \(B\) is a normal subgroup of \(G\), then \(G\) is solvable.