Qualifying Exam   Algebra   Spring 1991

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 $\cap$ A is a normal subgroup of H .

(b)

Prove that either H $\subseteq$ A or H/(H $\cap$ 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$\to$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 $\subseteq$ 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 $\cap$ 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
$$\begin{align*} 0 \longrightarrow S \overset {\alpha} {\longrightarrow} M \overset {\pi} {\longrightarrow} S \longrightarrow 0\end{align*}$$
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 $\alpha$(S) and M . (Hint: if 0$\ne$N $\subset$ M and $\alpha$(S) $\cap$ N = 0 , show that there is an isomorphism $\sigma$ : S$\to$N such that $\pi$$\sigma$ = 1_S .)

7.

Suppose that F is a Galois extension of $\mathbb {Q}$ with [F : $\mathbb {Q}$] = 25 . What possible groups can occur as the Galois group of F over $\mathbb {Q}$ ? In all cases, describe the intermediate fields between F and $\mathbb {Q}$ is in terms of inclusion and dimension over $\mathbb {Q}$ . Which intermediate fields are Galois over $\mathbb {Q}$ ?

8.

Recall that a group G of permutations of a set S is called transitive if given s,t $\in$ S , then there exists $\sigma$ $\in$ G such that $\sigma$(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) .