Math 6320 • Algebra II

Let \(n\) and \(m\) be two relatively prime natural numbers. Let \(N = nm\). Let \((G,+)\) be an abelian group such that \(0 = Ng = g+\cdots +g\) (\(N\) summands) for every \(g \in G\). Let \(G_n=\{g\in G \mid ng=0\}\) and \(G_m=\{g\in G \mid mg=0\}\). Then \(G_n\) and \(G_m\) are subgroups of \(G\). (You do not need to verify that.) Prove that \(G\) is isomorphic to \(G_n \oplus G_m\).