Algebra Prelim, Spring 1992
Answer all questions
- 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 .
- Prove that any group of order p 2q is solvable, where pq are primes. (Hint: consider separately the cases p > q and p < q ).
- List all groups of order 6 (up to isomorphism), and prove
that they are the only ones.
- If A and B are finitely generated abelian groups with A A isomorphic to B B , prove that A and B are
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 .
- 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 = .
- Let A be an abelian group and let m > 1 be an integer.
A /m A/mA .
- 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] .
- 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