Just to mention that, although the classification of finite abelian groups really isn't that hard, Dirichlet's theorem on primes in arithmetic progressions, on the other hand, is a pioneering and deep theorem that introduces several new ideas and was probably the first proof in analytic number theory. So I wonder if the existence of any finite group up to isomorphism as a multiplicative subsemigroup of at least one ring Z/N (N in Z+) can be proved by more elementary methods than an appeal to Dirichlet. —Dan On Apr 24, 2016, at 9:59 PM, W. Edwin Clark <wclark@mail.usf.edu> wrote: In fact I think that it is true that every finite abelian group is isomorphic to a subgroup of the group of units of Z/(n) for n a product of distinct primes. . . . . . . This follows from Dirichlet's theorem_on_arithmetic_progressions <https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressio... <https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions>> On Apr 24, 2016, at 9:54 PM, Andy Latto <andy.latto@pobox.com> wrote: Your conjecture is not hard to prove, if you know the fundamental theorem of Abelian groups (every abelian group is the direct product of cyclic groups) and Dirichlet's theorem on primes in arithmetic progressions (if a and b are relatively prime, there are infinitely many primes of the form an + b).