Algebra Prelim Spring 1994

1. Let G be a group with exactly three elements of order two. Prove that G is not simple.

2. Let . Prove that G has a homomorphic image isomorphic to S3 (the symmetric group of degree 3), but is not isomorphic to S3.

3. Let R be a PID which is not a field, and let M be a finitely generated R-module which is not a torsion module.

(i) Prove that the R-module R is isomorphic to a proper submodule of itself.

(ii) Prove that M is isomorphic to a proper submodule of itself.

4. Let R be a ring, and let A,B,C be R-modules.

(i) Prove that as abelian groups.

(ii) Prove that is not isomorphic to .

5. Let R be a commutative Noetherian ring.

(i) If S is a multiplicative subset of R, prove that S-1R is Noetherian.

(ii) Prove that R[[X-1,X]] (the Laurent Series ring in X) is a Noetherian ring (you may assume that the power series ring R[[X]] is Noetherian).

6. (i)  Prove that X4 + X3 +X2+X+1 is irreducible in . (Set Y=X-1.)

(ii) Let , and suppose no eigenvalue of A is equal to 1. Prove that if A5 = I, then 4|n (where I denotes the identity matrix).