Algebra Prelim Spring 1994
Answer All Problems

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).

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