**Algebra Prelim, Fall 1992**

Answer all questions

- Prove that any group of order 765 is abelian.
- Let
*G*be a finite group and let*N*be a normal subgroup of*G*. Prove that*G*is solvable if and only if both*N*and*G*/*N*are solvable. -
- Prove that if
*G*is a subgroup of*S*_{n}, then either or . - Prove that if and
*G*is a normal subgroup of*S*_{n}, then ,*A*_{n}or*S*_{n}. - Prove that if , then
*S*_{n}has no subgroup of index 3.

- Prove that if
- Let
*f*(*x*) be an irreducible polynomial over with splitting field*K*. If the Galois group of is abelian, prove that for any roots of*f*(*x*) in*K*, we have . - Let
*K*be a field and be a separable irreducible polynomial of degree 4, and let*E*be a splitting field for*f*(*x*) over*K*. If is a root of*f*(*x*) and , prove that there exists a subfield*F*of*L*with [*F*:*K*] = 2 if and only if the Galois group of*E*/*K*is not isomorphic to either*A*_{4}or*S*_{4}. -
- If
*G*,*H*and*K*are finitely generated abelian groups with , prove that . - Give an example to show that part (a) is false if
*G*is not finitely generated.

- If
- Let
*R*be an integral domain. A nonzero element of*R*is a*prime*if implies that either or . A nonzero element is*irreducible*if implies that either*a*or*b*is a unit.- Prove that every prime is irreducible.
- If
*R*is a UFD, prove that every irreducible is prime.

- Let
*R*be a commutative ring with a 1, and let*M*be a cyclic*R*-module.- Prove that
*M*is isomophic to*R*/*I*for some ideal*I*of*R*. - If
*N*is any*R*-module, prove that is isomorphic to*N*/*IN*for some ideal*I*of*R*.

- Prove that

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Tue Jul 30 17:01:19 EDT 1996