Review of Group Theory: The Structure Theorem For Finite Abelian Groups[Almost](Pt. I)
Point of post: In this post we discuss an prove a weak version of the structure theorem for finite abelian groups. Namely, for any abelian group there exists primes and integers such that and that this representation is unique.
It’s always been amazing how much simpler the theory for finite abelian groups has been then that of non-abelian groups. I mean, ostensibly the commutativity of the binary operation does seem like it would make things simpler, but really the theory of non-abelian groups is on another whole level. In this post we shall quantify how big of a difference it is by showing that every abelian group is just a direct product of cyclic groups whose orders are powers of primes. And that up to permutation of product factors this decomposition is unique. The difference between this and the case for non-abelian groups. In fact, the only similar theorem which says roughly “you only need to study this family of groups because the rest of the theory you care about is secretly hidden within” is Cayley’s theorem which makes the ‘parent’ family of groups the symmetric groups–which are not by any standard as nice as the direct product of cyclic groups whose orders are powers of primes.
Remark: The almost part is because I will only prove the existence part of the proof and not the uniqueness. The reason for this is that the uniqueness proof is my interpretation of a clever proof of this theorem given in Barry Simon’s book. The uniqueness part he gives I am not nearly as fond of. And, since I know a proof and it is the proof that is in all the books (Rotman, Dummit and Foote, Lang, Grillet, etc.) there is no use copying it here.
Structure Theorem for Finite Abelian Groups
We begin a lemma which can be seen as a corollary from a previous theorem:
Lemma 1: Let where . Then,
Proof: This follow immediately from our previous theorem that if then applied multiple times.
Corollary: If are primes and then:
We next show then that any abelian group is isomorphic to a group of the form :
Lemma 2: Let be a finite abelian group, then there exists such that
Proof: We say that a group is -generatable if there exists such that generates. Clearly every finite group is -generatable since . We induct on . For this is trivial since by definition every -generatable group is cyclic and thus isomorphic to .
Suppose then that every -generatable group can be written as the direct product of cyclic groups and let be -generatable. For any and any -tuple define by
It’s clear via the fact that is abelian that is a homomorphism. Let then . If we assume that generates then by definition is an epimorphism and so by the first isomorphism theorem we have . Now, let be the first canonical projection onto the first coordinate given by and define
and let and find which generates such that for some . We now claim that if then . Indeed, by the division algorithm we have that where and
and so and thus by construction we must have that from where follows. It follows then that if that for some and so
but it’s easy to show that . What we claim though next is that for . Indeed, by the division algorithm write for and let . Clearly then generate and since we must have for otherwise if, say , then and contradicting the minimality of . Thus, . Note though that if we define and then by we have already proven that . Noting then that is trivial it easily follows that . Note though that by prior observation we have that and so evidently . Moreover, it’s clear that and that if then . Thus
But since is, by construction, -generatable it is by the induction hypothesis isomorphic to a direct product of cyclic groups and so evidently from so is . The induction is complete.
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.