Math 561 H Fall 2011 Homework 3 Solutions Drew Armstrong
image of ’is the (additive) cyclic subgroup hai Z=nZ. Then since ’is surjective we must have hai= Z=nZ. By the Lemma, this happens if and only if aand nare coprime, i.e. a2(Z=nZ) , in which case the map ’(x) = axis also invertible with inverse ’ 1(x) = a 1x.