I am trying to do all the exercises in Rotman (3rd ed.). It's an integral part of his exposition to do many of them, but this one is optional. I don't get it... (problem 2.83 in the book).

G=H x K, N is normal in G. Prove: Either N is abelian OR N intersects one of the factors non-trivially.

Suppose N intersects H in the identity AND N intersects K in the identity (i.e., trivially in both cases). The proposition is then that N must be abelian.

I think we have the following fact: Every element of N must be of the form n=hk with neither h,k the identity unless both are. From this, we can deduce that every first component, h, has the same order, m, as every second component, k.

h^m=1

k^m=1

But does this make N abelian? No.

Note: Presumably, Sylow theory is not used to solve this, since it precedes that theory.

Any help will be greatly appreciated, and apologies in advance if I'm being dense.