Solving linear equations over finitely generated abelian groups

TL;DR

This paper explores algorithms for solving linear equations over finitely generated abelian groups, focusing on membership testing, pre-image computation, and kernel determination, which are fundamental for understanding homomorphisms in algebraic structures.

Contribution

It introduces and analyzes various algorithms for solving linear equations over finitely generated abelian groups, providing effective methods for key computational problems.

Findings

Algorithms for membership testing are effective.

Pre-image computation methods are developed.

Kernel determination techniques are presented.

Abstract

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 $\varphi$.