Linear equations over finite abelian groups
One of the oldest problems in algebra is the solution of a system of linear equations over certain domains. The Gaussian elimination algorithm provides an effective solution over fields. The Smith normal form algorithm yields a method over the integers. Here we consider another variation of this theme. Let A be a finite (additive) abelian group, let αi,j ∈ End(A) for 1 ≤ i ≤ n and 1 ≤ j ≤ m and let b1, . . . , bm ∈ A. Then we want to determine the set S of all (x1, . . . , xn) ∈ A n with
Solving linear equations over finitely generated abelian groups
We discuss various methods and their effectiveness for solving linear equations over finitely generated abelian groups. More precisely, if $\varphi\colon G\to H$ is a homomorphism of finitely generated abelian groups and $b\in H$, we discuss various algorithms for checking whether $b\in \im\varphi$ holds and if so, for computing a pre-image of $b$ in $G$ together with the kernel of $\varphi$.
On Systems of Linear Diophantine Equations
Introduction Something happened to me recently I would wager has happened to many who read this note. Teaching a new topic, you cannot understand one of the proofs. Your first attempt to fill the gap fails. You look through your books for an answer. Next, you ask colleagues, go to the library, maybe even use the interlibrary loan. All in vain. Then it strikes you that, in fact, you cannot answer an even more basic and seemingly more interesting question. You peruse the books again. They seem to have answers to thousands of strange questions, but not to yours (the most natural one!). At the same time you cannot believe that your question could have been overlooked by generations of mathematicians. Days pass; the agony continues. Then one day, some way or other, you find the answer. In my case the answer was in a book I already owned. It followed from a theorem I had known for a long time, but I had never thought of this particular application. I must admit, indeed, that this theorem appeared in almost every book I had checked, but never with a pointer to this particular application, even as an exercise. Were the authors unaware of the application? Or did it seem to obvious to mention? In any case, here is the story.