Suppose $X,Y,Z$ are finite abelian groups with $X \times Y \cong X \times Z$. How to show that $Y\cong Z$?
If we assume that we can decompose $Y,Z$ into cyclic groups that are powers of primes, I believe the problem becomes trivial. But suppose that we only know that we can decompose finite abelian groups as $\mathbb{Z}_{a_1} \times \dots \times \mathbb{Z}_{a_n}$ where $a_1|\dots |a_n$. How to proceed from there?