Proving Cauchy's Theorem with induction

[beachbum300]

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 without 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

Homework Equations

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?

 

[Focus]

Take any non-identity element of G, say a, and use $n=[G:<a>]|<a>|$.