Algebra Prelim, Fall 1992

1. Prove that any group of order 765 is abelian.
2. Let G be a finite group and let N be a normal subgroup of G. Prove that G is solvable if and only if both N and G/N are solvable.
1. Prove that if G is a subgroup of Sn, then either or .
2. Prove that if and G is a normal subgroup of Sn, then , An or Sn.
3. Prove that if , then Sn has no subgroup of index 3.
3. Let f(x) be an irreducible polynomial over with splitting field K. If the Galois group of is abelian, prove that for any roots of f(x) in K, we have .
4. Let K be a field and be a separable irreducible polynomial of degree 4, and let E be a splitting field for f(x) over K. If is a root of f(x) and , prove that there exists a subfield F of L with [F:K] = 2 if and only if the Galois group of E/K is not isomorphic to either A4 or S4.
1. If G, H and K are finitely generated abelian groups with , prove that .
2. Give an example to show that part (a) is false if G is not finitely generated.
5. Let R be an integral domain. A nonzero element of R is a prime if implies that either or . A nonzero element is irreducible if implies that either a or b is a unit.
1. Prove that every prime is irreducible.
2. If R is a UFD, prove that every irreducible is prime.
6. Let R be a commutative ring with a 1, and let M be a cyclic R-module.
1. Prove that M is isomophic to R/I for some ideal I of R.
2. If N is any R-module, prove that is isomorphic to N/IN for some ideal I of R.