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Finite group of even order has elements of order 2

  1. Aug 25, 2014 #1
    [The homework format does not appear on mobile]

    Problem: Show that a finite group of even order has elements of order 2

    Attempt:
    The book gives a suggested approach that lead me to write the most round about, ugly proof I've ever written.

    Can't I just say:
    1.) If G has even order, G/{1} has odd cardinality.
    2.) Assume that no elements of G/{1} has order 2.
    3.) Then for each x in G/{1}, x^-1 is a distinct element of G/{1}.
    4.) Then G/{1} has even cardinality.
    Contradiction
    5.) Therefore, there exists a y in G/{1} such that y = y^-1, and thus has order 2.
     
  2. jcsd
  3. Aug 25, 2014 #2

    pasmith

    User Avatar
    Homework Helper

    That is the standard method of showing that a finite group of even order has an element of order 2.
     
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