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Proving Cauchy's Theorem with induction

  1. Mar 25, 2009 #1
    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?
     
  2. jcsd
  3. Mar 25, 2009 #2
    Take any non-identity element of G, say a, and use [itex]n=[G:<a>]|<a>|[/itex].
     
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