I have five problems from my text that I did exactly the same way. Thus I have reason to believe that I am doing something wrong.
The problem is showing that two groups are not isomorphic. To give you the flavor of what I am doing consider the following:
It is clear to me that we can construct a function that is surjective. The original reason I claimed that the function can't be injective is because is a two parameter group and is only a one parameter group, so we cannot construct a 1 - 1 function, therefore the groups cannot be isomorphic. I don't see this reasoning to be correct any more as I've been applying this over and over in the exercises. (I don't see any reason the text would hammer the point over and over if the reasoning is this "trivial.")Prove that the additive groups and are not isomorphic.
So how would I show that these groups are not isomorphic?