We caracterize a (affine) sub-group of Zn by a polynomial ideal and standard (Gröbner) basis. We show how to use these standard basis to solve directly many algorithmic problems on (affine) sub-groups and their non-negative parts : * the membership problem * the triviality problem * find the smallest non zero vector * find the minimal non-negative elements * solve linear diohantine systems (Ax = 0, x>=0), (Ax = b, x>=0), Ax <= b on integers, and find smallest solutions Finally we conjecture other properties of these standard basis