Math 548: Assignment Due February 10th

(a) Let d = (m,n). Then notice that d · nd = n ≡ 0 ∈ Z/n, so nd is in the kernel of the map ·m. Now let k ∈ Z/n satisfy mk = 0 ∈ Z/n. Then we have mk = c′n for some c′ ∈ Z. We can write n = d · nd , and so we have mk = cdd . Dividing both sides by d, we find m d k = c ′ n d where m d ∈ Z. But md is relatively prime to n, and so letting k = ( m d )−1 c′ nd , we see that k ∈ {n/d} and we are done. (b) Let G be a cyclic group, and let H < G be a subgroup. Note that G is abelian, so that H is normal and hence we can consider G/H as a group. Let G =< g >, and let h ∈ G/H, where h ∈ G is a representative of h. We can write h = ng. If n ≥ 0, we then have h = g + g + · · · + g, so that h = g + g + · · · + g, i.e. h = ng. If n ≤ 0, we have similarly h = −g − g − · · · − g, so that h = −g +−g + · · ·+−g = −g − g − · · · − g, and hence h = ng. Thus G/H =< g >.