Efficient algorithms for the basis of finite Abelian groups

TL;DR

This paper presents efficient algorithms with linear and near-linear time complexities for computing bases of finite abelian groups, improving computational methods in algebra.

Contribution

Introduces new algorithms for basis computation in finite abelian groups with optimal and near-optimal time complexities.

Findings

O(N) time algorithm for basis computation in finite abelian groups

Algorithm for basis from generating system with complexity depending on prime divisors

Special case algorithm for cyclic groups with near-linear time complexity

Abstract

Let $G$ be a finite abelian group $G$ with $N$ elements. In this paper we give a O(N) time algorithm for computing a basis of $G$. Furthermore, we obtain an algorithm for computing a basis from a generating system of $G$ with $M$ elements having time complexity $O(M\sum_{p|N} e(p)\lceil p^{1/2}\rceil^{\mu(p)})$, where $p$ runs over all the prime divisors of $N$, and $p^{e(p)}$, $\mu(p)$ are the exponent and the number of cyclic groups which are direct factors of the $p$-primary component of $G$, respectively. In case where $G$ is a cyclic group having a generating system with $M$ elements, a $O(MN^{\epsilon})$ time algorithm for the computation of a basis of $G$ is obtained.