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    binary operation

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    Prove that any non empty set G with a binary operation satisfying the following requirements is a group:
    a) if x, y in G then x.y in G
    b) for all x,y,z in G, (x.y).z = x.(y.z)
    c) for all a,b in G, the equations x.a =b and a.y =b both have solutions.

    Hint: 1st show that if a in G then a has a right and left identity and be sure to show they are the same.
    2ed show that the identity in step #1 works for any b in G
    Note: 1st and 2ed steps can be reversed
    3ed show that for any a in G a has a left and right inverse and they are the same.

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    https://brainmass.com/math/discrete-math/discrete-math-binary-operation-346348

    Solution Preview

    Dear Student
    Given that for all a,b in G, the equations x.a =b and a.y =b both have solutions, hence we should have a solution for the equation x.a=a too.
    This implies that there is an element 'e' (the solution for the above equation for x=e) such that e.a=a, ...

    Solution Summary

    This solution shows how to prove that any non empty set G with a binary operation, satisfying the following requirements is a group.

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