Efficient algorithms for the basis of finite Abelian groups

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 ∑ p|N e(p)⌈p 1/2⌉μ(p)), where p runs over all the prime divisors of N , and pe(p), μ(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 ǫ) time algorithm for the computation of a basis of G is obtained.