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**1. Homework Statement**

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. Homework 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?