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Math Help - Abelian Group proof

  1. #1
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    Abelian Group proof

    Suppose a,b e G, where G is an abelian group under +. Show -(a + b) = (-a) + (-b).

    Thanks for any help.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by jzellt View Post
    Suppose a,b e G, where G is an abelian group under +. Show -(a + b) = (-a) + (-b).

    Thanks for any help.
    By a property of inverses, -(a+b) = (-b) + (-a). Since a,b\in G and G is Abelian, then (-b)+(-a)=(-a) + (-b)
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    How do we know that -(a + b) = (-b) + (-a)?

    Isn't that something I would have to show while giving a solid proof?
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by jzellt View Post
    How do we know that -(a + b) = (-b) + (-a)?

    Isn't that something I would have to show while giving a solid proof?
    You need to keep in mind that if \left(G,*\right) is a group a,b\in G, then \left(a*b\right)^{-1}=b^{-1}*a^{-1}. In additive notation, this would be the same as -(a*b)=(-b)*(-a), where * is a binary operation. In this case, that binary operation is +.

    Does this clarify things?
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