Problem 1: Suppose G is a group with the property that agb=cgd always implies that ab=cd. Prove that G is abelian.
What does this question mean? Is g a special element of G, or any old element? Suppose g is the identity (in which case ab=cd anyway). How does one prove this?
Problem 2: Suppose G is a finite abelian group in which no non-identity is equal to its own inverse. Determine the product of all the elements of G.
How can I do the entire table? It could have thousands (literally!) of entries (or more). How do I know what a time b equals? I don't get it.