Qualifying Examination Algebra Fall 1995

#1. Suppose that G is a group of order (35)3. Show that G has normal Sylow 5- and 7-subgroups. Also show that G has a normal subgroup of order 25.

#2. Suppose that G is an abelian group isomorphic to . Let H be a subgroup of G of order 27. Up to isomorphism, describe H.

#3. Let be a group epimorphism with G finite, let , and let C denote the centralizer of f(g) in H.

a). Prove that if D is a conjugacy class in f-1(C), then f(D) is a conjugacy class in C.

b). Prove that the order of the conjugacy class of g in f-1(C) is at most .

c). Prove that the order of the centralizer of g in G is at least |C|.

#4. Suppose that R is a unique factorization domain which is NOT a principal ideal domain.

a). Show that R must have at least two (nonassociate) prime elements.

b). Show that R must have a nonprincipal maximal ideal.

#5. Let R be a ring with a 1 and suppose that X is an R-module and N is a submodule of an R-module M. Let denote the inclusion map and let be the R-module homomorphism . Prove that if

is a commutative diagram for some R-homomorphism f, then M is isomorphic to .

#6. Suppose that E is a Galois field extension of F with [E:F]=pn for some prime p and positive integer n. Show that there are intermediate fields so that [Ki:Ki-1] = p and Ki is Galois over F for i =1, ... , n.

#7. Prove that there is no finite field which is algebraically closed.