**Algebra Preliminary Exam, Fall 1987**

Instructions: do all problems

- Let
*G*be a group with 56 elements. Prove that*G*is not simple. - Let
*G*be a group and let . Prove that . Now suppose that*G*=*HA*where*H*and*A*are subgroups and*A*is abelian. Prove that there exists such that . Deduce that if*G*is nonabelian simple, then for all . - Let
*G*be the group with the operation multiplication. Define by . Prove that is a group homomorphism, , and . Suppose for some positive integer . Is . - If
*H*is a subgroup of the group*G*, let denote the normalizer of*H*in*G*. Suppose*G*is a finite group and*P*is a Sylow*p*-subgroup of*G*. Prove that . - Let
*R*be a commutative ring. If*I*and*J*are ideals of*R*, define . Prove that (*I*:*J*) is an ideal of*R*.Now suppose , ,

*K*= (*I*:(*a*)) and*R*/*I*is a domain. Prove that*K*=*I*and*aK*=*I*. Deduce that . Does the final assertion remain true if the hypothesis is dropped? ((*a*) denotes the ideal generated by*a*.) - Let
*K*be a field. Prove that*K*[*X*] has infinitely many irreducible polynomials, no two of which are associates. (Consider*p*_{1}*p*_{2}...*p*_{n}+ 1). Suppose now , . Prove that there exists a homomorphism from*K*[*X*] to a domain with nonzero kernel such that . - Let
*R*be a ring, let*M*be an*R*-module, and let be an*R*-module homomorphism. Prove that is a submodule of*M*.Now suppose every submodule of

*M*is finitely generated. Prove there exists an integer*n*such that . Deduce that if is onto, then is an isomorphism. - Let
*V*be a vector space over and let be a linear transformation. Describe how*V*can be made into a -module via*T*.Now let be a basis for

*V*and suppose*T*(*e*_{1}) = -*e*_{1}+ 2*e*_{2},*T*(*e*_{2}) = -2*e*_{1}+3*e*_{2},*T*(*e*_{3}) = -2*e*_{1}+2*e*_{2}+*e*_{3}. Find the Jordan canonical form for the matrix of*T*. Hence find the isomorphism type of*V*(as a -module) as a direct sum of primary cyclic modules. Does there exist a -module homomorphism of onto*V*?

Tue Apr 15 13:38:47 EDT 1997