Algebra Prelim, Spring 1992

Answer all questions

1.

(a)

Let G be a finite group of order m , and let p be the smallest prime which divides m . Prove that if H is a subgroup of index p , then H is a normal subgroup of G .

(b)

Prove that any group of order p ²q is solvable, where p$\ne$q are primes. (Hint: consider separately the cases p > q and p < q ).

2.

List all groups of order 6 (up to isomorphism), and prove that they are the only ones.

3.

If A and B are finitely generated abelian groups with A $\oplus$ A isomorphic to B $\oplus$ B , prove that A and B are isomorphic.

4.

Suppose 0 $\longrightarrow$ A$\overset {f}$ $\longrightarrow$ B$\overset {g}$ $\longrightarrow$ C $\longrightarrow$ 0 is a short exact sequence of modules over a ring R . Prove that if the sequence splits, i.e. there is an R -module homomorphism h : C$\to$B such that gh = 1_C , then B $\cong$ A $\oplus$ C .

5.

Let R be a commutative ring with a 1. If S is a multiplicative set (i.e. x,y $\in$ S $\Rightarrow$ xy $\in$ S ) containing 1, but not 0, prove there exists a prime ideal of P of R with P $\cap$ S = $\emptyset$ .

6.

Let A be an abelian group and let m > 1 be an integer. Prove that A $\otimes$ $\mathbb {Z}$/m$\mathbb {Z}$ $\cong$ A/mA .

7.

Let K/k be a Galois extension of fields and let f (x) $\in$ k[x] be an irreducible polynomial which has a root in K . Prove that f (x) splits into linear factors in K[x] .

8.

Let K be a finite Galois extension of $\mathbb {Q}$ with Galois group isomorphic to A₄ . For each divisor d of 12, how many subfields L of K have [K : L] = d ? In each case give the isomorphism class of $\Gal$(K/L) , and state whether or not L/$\mathbb {Q}$ is a Galois extension. (Recall A₄ is a counter-example to the converse of Lagrange's theorem.)