- 1.
- Suppose that
*A*,*H*are normal subgroups of a group*G*such that*G*/*A*is a simple group of order*n*.- (a)
- Prove that
*H**A*is a normal subgroup of*H*. - (b)
- Prove that either
*H**A*or*H*/(*H**A*) is a simple group of order*n*. (Hint: use an isomorphim theorem.)

- 2.
- (a)
- Prove that a group of order 100 cannot be simple.
- (b)
- Describe all abelian groups of order 100 up to isomorphism.
- (c)
- Either show that every group of order 100 is abelian, or exhibit a nonabelian example.

- 3.
- Let
*G*be a group and let*f*:*G**H*be a group homomorphism. Prove that if*H*is a solvable group and if ker (*f*) is abelian, then*G*is a solvable. - 4.
- Let
*R*be a PID.- (a)
- Prove that the intersection of two nonzero maximal ideals cannot be zero.
- (b)
- Assume that
*R*contains an infinite number of maximal ideals. Show that the intersection of all the nonzero maximal ideals of*R*equals zero.

- 5.
- Let
*R**S*be rings with a 1 such that*S*/*R*is a free left*R*-module. Prove that if*L*is a left ideal of*R*, then*LS**R*=*L*. (Hint: write*S*as a direct sum of*R*-modules.) - 6.
- Let
*R*be a ring with 1. A nonzero left*R*-module*S*is simple if 0 and*S*are the only submodules of*S*. Let

be a short exact sequence of*R*-modules which is*not*split, and such that*S*is a simple*R*-module. Show that the only nonzero submodules of*M*are (*S*) and*M*. (Hint: if 0*N**M*and (*S*)*N*= 0 , show that there is an isomorphism :*S**N*such that = 1_{S}.) - 7.
- Suppose that
*F*is a Galois extension of with [*F*: ] = 25 . What possible groups can occur as the Galois group of*F*over ? In all cases, describe the intermediate fields between*F*and is in terms of inclusion and dimension over . Which intermediate fields are Galois over ? - 8.
- Recall that a group
*G*of permutations of a set*S*is called*transitive*if given*s*,*t**S*, then there exists*G*such that (*s*) =*t*. Let*f*(*x*) be a separable polynomial in*K*[*x*] and let*F*be a splitting field of*f*(*x*) over*K*. Prove that*f*(*x*) is irreducible over*K*if and only if the Galois group of*F*over*K*is a transitive subgroup when viewed as permutations of the roots of*f*(*x*) .