sylow problem (Hernstein)

If G is of order p^(2)*q,p,q primes, prove that either a p-Sylow sub-group or a q-Sylow subgroup of G must be normal in G. this is prob 14 in sec 2.12 of Hernstein's Topics of Alg.

Abelian group problem

Let G be an Abelian group such that the operation on G is denoted additively. Show that {a is an element of G| 2a = 0} os a subgroup of G. Compute the subgroup for G =13.

Rings that are not Isomorphic

I need to prove that the rings 2Z and 3Z are not isomorphic. Using the method used here, I must also be able to show that the rings R (set of reals) and C (set of complex) are not isomorphic.

Nilpotent Elements of Commutative Rings

I need to prove the following: An element a of a ring R is nilpotent if a^n=0 for some n in Z+. Show that if a and b are nilpotent elements of a commutative ring, then a + b are also nilpotent.

Rings with Unity that forms a group

I need to prove the following: Show that if U is the collection of all units in a ring with unity, then is a group. A reminder was given to make sure to show that U is closed under multiplication. Thanks for the Help.