Algebra Preliminary Exam, Fall 1980

1. Suppose that for each prime integer p dividing the order of a finite group G, there is a subgroup of index p. Prove that G cannot be a nonabelian simple group.
2. A subgroup of a finite group is ``p-local" if it is the normalizer of some Sylow p-subgroup. Show that the number of p-local subgroups of a group is congruent to 1 modulo p.
3. Characterize (with proof) all finitely generated -modules with the property that each submodule is a direct summand. ( denotes the field of rational numbers.)
4. Let R and S be local Noetherian integral domains with maximal ideals M and N respectively. Assume that and that S is a finitely generated R-module. If there exists a proper ideal I of R such that and the canonical image of R/I in S/IS equals S/IS, then prove that R=S.
5. Let R be an integral domain. For , define . Let be a set of prime ideals of R with the property that if and , then for some . Prove that . ( denotes the localization at .)
6. If G is a finite group, prove that there exist fields K and L such that L is a Galois extension of K with Galois group isomorphic to G.
7. Show that for each positive integer n, there is a polynomial such that for each matrix A with complex entries,

   

   ( denotes the field of complex numbers and denotes the trace of A.)

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