Algebra Prelim, Spring 1992

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 pq 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 A isomorphic to B B , prove that A and B are isomorphic.

4.
Suppose 0 A B C 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 : CB such that gh = 1C , then B A C .

5.
Let R be a commutative ring with a 1. If S is a multiplicative set (i.e. x,y S xy S ) containing 1, but not 0, prove there exists a prime ideal of P of R with P S = .

6.
Let A be an abelian group and let m > 1 be an integer. Prove that A /m A/mA .

7.
Let K/k be a Galois extension of fields and let f (x) 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 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 (K/L) , and state whether or not L/ is a Galois extension. (Recall A4 is a counter-example to the converse of Lagrange's theorem.)

Peter Linnell
8/13/1997