So I'm working through Koblitz'z "a course in number theory and cryptography" when I came across his proof that every finite field has a generator (ie, There is an element such that the multiplicative group generated by this element are all the units of the field). This implies that the multiplicative group of any finite field is cyclic. So this got me thinking about a number of things. The first being, since the proof only relies on multiplicative commutativity and finiteness, is every finite abelian group cyclic? If not, then there must be a restriction on what finite abelian groups can possibly be the multiplicative group of some finite field. The proof is on page 34 for anyone that has a copy and wants to shed some light on the matter. I'm really just trying to understand what additional structure multiplicative groups of finite fields are endowed with that arbitrary finite abelian groups lack.
Thanks! -wrote using ipad; beware of typos!