Qualifying Exam   Algebra   Spring 1991

Suppose that A,H are normal subgroups of a group G such that G/A is a simple group of order n .
Prove that H $\cap$ A is a normal subgroup of H .
Prove that either H $\subseteq$ A or H/(H $\cap$ A) is a simple group of order n . (Hint: use an isomorphim theorem.)
Prove that a group of order 100 cannot be simple.
Describe all abelian groups of order 100 up to isomorphism.

Either show that every group of order 100 is abelian, or exhibit a nonabelian example.

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.

Let R be a PID.
Prove that the intersection of two nonzero maximal ideals cannot be zero.
Assume that R contains an infinite number of maximal ideals. Show that the intersection of all the nonzero maximal ideals of R equals zero.

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.)

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
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$ = 1S .)

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}
$ ?

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) .


Peter Linnell