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Math Help - Groups; solution to quadratic.

  1. #1
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    Groups; solution to quadratic.

    I'm new to groups and trying to answer the following question.

    Let G be any group. Show that the identity e is the unique solution of the equation x^2=x

    So how do I proceed?
    A group, as far as my primitive understanding goes, is a set governed by a binary operation, producing local inbred offspring; then we have the given axioms: closure, associativity, identity and invertibility.

    So we assume G is a group with a binary operation x*x=x^2or is it x^2-1 ?

    Then what?

    we know x*e=xe-1=x
    and e*e=e^2-1=e

    Does this help?

    What is the best and most comprehensive text on groups that starts at the absolute beginning?
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  2. #2
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    You know that elements in a group are always invertible, so what happens when you multiply the identity x^2=x by x^{-1}.

    As for texts in groups, Rotman's An Introduction to the theory of groups is very good in my opinion.
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  3. #3
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    If we let the binary operation be x*x=x^2 and we know that  x*e=x then (given x^2=x) x*x=x*e therefore x=e

    How do we show it is a unique solution?
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