Linear and Sublinear Time Algorithms for Basis of Abelian Groups

Abstract

AbstractIt is well known that every finite abelian group G can be represented as a direct product of cyclic groups: G ≅ G 1×G 2× ⋯ ×G t , where each G i is a cyclic group of order p j for some prime p and integer j ≥ 1. If a i generates the cyclic group of G i , i = 1,2, ⋯ , t, then the elements a 1,a 2, ⋯ , a t are called a basis of G. We show a randomized algorithm such that given a set of generators M = {x 1, ⋯ , x k } for an abelian group G and the prime factorization of order ord(x i ) (i = 1, ⋯ , k), it computes a basis of G in \(O(|M|(\log n)^2+\sum_{i=1}^t n_ip_i^{n_i/2})\) time, where n = |G| has prime factorization \(p_1^{n_1}p_2^{n_2}\cdots p_t^{n_t}\) (which is not a part of input). This generalizes Buchmann and Schmidt’s algorithm that takes \(O(|M|\sqrt{|G|})\) time. In another model, all elements in an abelian group are put into a list as a part of input. We obtain an O(n) time deterministic algorithm and a subliner time randomized algorithm for computing a basis of an abelian group.KeywordsAbelian GroupPrime FactorizationCyclic GroupRandom ElementDeterministic AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.