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Math Help - Exponent Laws for Groups

  1. #1
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    Exponent Laws for Groups

    How do you prove the following exponent laws for groups?

    let x be an element of group G. Then for all integers m and n:

    (i) x^-n = (x^-1)^n
    (ii) x^m . x^n = x^(m+n)
    (iii) x^m . x^n = x^n . x^m
    (iv) (x^m)^n = x^(mn)

    They seem self-evident, but I'm not quite sure how to prove them, especially when m and n are negative integers.
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  2. #2
    Senior Member roninpro's Avatar
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    You need to write down the definition. For n\geq 1, the expression x^n is defined to be x\cdot x\cdots x ( n times). We also define x^0=e and x^{-n} to be the inverse of x^n.

    Try using these to prove your statements.
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  3. #3
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    I think I can prove statements (ii), (iii) and (iv) for all non-negative integers m and n using induction on either m or n. However, I don't know how to prove the statements for negative integers.
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Alfie View Post
    How do you prove the following exponent laws for groups?

    let x be an element of group G. Then for all integers m and n:

    (i) x^-n = (x^-1)^n
    (ii) x^m . x^n = x^(m+n)
    (iii) x^m . x^n = x^n . x^m
    (iv) (x^m)^n = x^(mn)

    They seem self-evident, but I'm not quite sure how to prove them, especially when m and n are negative integers.
    Personally, I would show that \langle x\rangle\cong \mathbb{Z}/n\mathbb{Z} for some n \in \mathbb{Z}, or \langle x\rangle \cong \mathbb{Z} in a `natural' way. You can then use this isomorphism to prove your result (you need to work out what I mean by `natural' (that is, find the isomorphism), because this is really the thing you use; you can't use the fact that the are isomorphic, you use the actual isomorphism).
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