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Math Help - Group of even order

  1. #1
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    Group of even order

    If G is a finite group of even order, show that there must be an element a not equal to e such that a = a^-1

    if I understand correctly, order of a group = number of elements and order of an element is the least integer m such that a^m = e.

    so if a does not equal e than a is not the identity element and e^1 =1 so e has order 1


    then if a = a^-1 then there is at least one element that is its own inverse, correct?

    and by definition, a*a^-1 =e so this is the same as a*a=e or a^2=e ?? so a is of order 2

    further, because e is an element that is it's own inverse then there's left an odd number of elements so at least one other element must be it's own inverse.
    Last edited by Jonroberts74; May 11th 2014 at 05:53 PM.
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  2. #2
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    Re: Group of even order

    Your last sentence correctly proves the result.
    Thanks from Jonroberts74
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