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Math Help - Order of elements in a group

  1. #1
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    Order of elements in a group

    Prove that in any group, an element and its inverse has the same order.

    My Proof:

    Let G be a group, and suppose that a \in G and a^{-1} \in G.

    By definitions, <a> = n such that  a^{n} = e and <a^{-1} = m such that  (a^{-1})^{m} = e

    thus,  a^{n} = a^{-m}

    Now, can I say that n must equal to m because the negative of inverse equals to the original power?
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  2. #2
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    Let a\in G have finite order n, i.e. a^n = 1 and n is the smallest such exponents.
    Thus,
    \underbrace{a\cdot ... \cdot a}_n=1
    Thus,
    (a\cdot ... \cdot a)^{-1} = 1
    Thus,
    a^{-1} \cdot ... \cdot a^{-1} = 1
    Thus,
    (a^{-1})^n = 1.
    Now it remains to show that n is smallest exponent for a^{-1}. Try showing that.
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  3. #3
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    Now, \underbrace{a^{-1} \cdot ... \cdot a^{-1}}_n=(a^{-1})^{n}=a^{-n}=(a^{n})^{-1}=(1)^{-1}=1

    Thus shows that (a^{-1})^{n}=e=a^{n}

    Q.E.D.

    Is that right?
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  4. #4
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    Quote Originally Posted by tttcomrader View Post
    Is that right?
    Almost. You need to show this is the smallest such exponent.
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