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Math Help - Groups problem...(Axioms)

  1. #1
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    Groups problem...(Axioms)

    Say S is a set of elements (finitely or infinetly). Suppose that S is closed under the operation *, and that the associative law holds. If a*e=a and there exists -a (the inverse element) such that a*(-a)=e for all a in S. Then e*a=a and (-a)*a=e and the uniqueness of e (the identity element) can be deducted.


    A little help with this is needed. Thanks in advance.
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  2. #2
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    Quote Originally Posted by Kichigo View Post
    Say S is a set of elements (finitely or infinetly). Suppose that S is closed under the operation *, and that the associative law holds. If a*e=a and there exists -a (the inverse element) such that a*(-a)=e for all a in S. Then e*a=a and (-a)*a=e and the uniqueness of e (the identity element) can be deducted.


    A little help with this is needed. Thanks in advance.

    First prove that in  S\,,\,\,x\cdot x=x\Longrightarrow x=e , and then prove that (a\cdot a^{-1})(a\cdot a^{-1})=\a\cdot a^{-1} (it seems to be that a^{-1} is more usual and clearer to denote the inverse of a than -a).
    Finally, use the above to prove e\cdot a=a

    Tonio
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