Algebra Preliminary Exam, Spring 1981

1. Prove that a group of order pⁿ, where p is a prime and , has a nontrivial center.
2. Let G be a group of order pq, where p and q are primes and p < q. Prove that G has only one subgroup of order q.
3. 1. Show that if H is a subgroup of order 12 in a group G of order 36, then H is a normal subgroup of G. (Map G to the automorphisms of the set of right cosets of H.) NOTE: there are examples of groups of order 36 with subgroups of order 12 which are not normal.
   2. Describe all abelian groups of order 36, up to isomorphism.
4. Prove that if p(x) is a polynomial irreducible over a field F, and if and are roots of p(x) in some extension field of F, then and are isomorphic. What happens if p(x) is reducible?
5. Let L be a normal extension of a field F, let , and let H be a subgroup of G. Prove that
   1. if for all , then K is a subfield of L;
   2. if K is normal over F, then H is a normal subgroup of G.
6. 1. Let R be a commutative principal ideal domain (PID) with a 1. If , show that a and b have a greatest common divisor (i.e. d divides a and b, and if g divides a and b, then g divides d).
   2. Show that is not a PID.
7. Let R be a commutative ring with identity, and let I,J be ideals in R. Let for .
   1. Show that IJ is an ideal of R.
   2. Show that if I+J = R, where , then (recall that R has a 1).
   3. Suppose further that R is a domain, and that for all ideals I,J in R. Prove that R is a field. (Take principal ideals.)

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