1. The problem statement, all variables and given/known data If p is a prime and p divides the order of G where G is a finite abelian group, then G has an element of order p. Prove with out using the Fundamenthal Theorem for Finitely Generated Abelian Groups. Hint 1: Induct on the order of G Hint 2: Note that G/H has a smaller order than G if H is non-trivial 2. Relevant equations 3. The attempt at a solution P(n): If the order of G=n and p divides the order of G, then G has an element of order p. Base Case: Consider n=1. Vacuously true. Consider n=2. Since 2 is prime and 2 divides 2, then G has an element of order 2 Consider n=3. Since 3 is prime and 3 divides 3, then G has an element of order 3. Inductive Step: Assume P(k) is true for 0<k<n where n>3. This is where I'm not sure of what to do next. Any suggestions?