7 problems.
- A normal \(p\)-subgroup lies in every Sylow \(p\)-subgroup, and in a maximal subgroup or with \(p\)-power index
- Stabilizers in an orbit are conjugate, conjugates of a proper subgroup do not cover \(G\), and a transitive action on two or more points has a fixed-point-free element
- Galois-ness of \(K/F\) and of \(L/K\) when \(L/F\) is Galois, and of \(L/F\) when \(K/F\) and \(L/K\) are Galois
- Fitting’s lemma: \(V=U\oplus W\) with \(\phi|_U\) nilpotent and \(\phi|_W\) nonsingular
- Cyclic vectors, the commutant, and equality of minimal and characteristic polynomials
- Maximal submodules are those with simple quotient, and the roots of unity have no maximal submodule
- A commutative ring with \(n\) zero-divisors has order at most \(n^2\), and the two rings with exactly two zero-divisors