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    A) Let G be a nonempty finite set equipped with an associative operation such that for all a,b,c,d in G:
    if ab = ac, then b = c and if bd = cd, then b = c.
    Prove that G is a group.

    b) Show that part (a) may be false if G is infinite

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    Solution Preview

    For a more readable version of the solution, please see the attached file "Solution.pdf".

    Note: Throughout this proof, associativity is implicitly used by not grouping the terms into brackets. If necessary, it can be explicitly shown (at the cost of a longer proof).

    Since G is finite, let G = {a_1, a_2, ... a_n}.
    Now consider the set S = {(a_1)(a_k) | k = 1, 2, ... n}.
    Since the operation is defined on G, (a_j)(a_k) is in G, for all a_j and a_k in G. Thus, the set S is a subset of G.
    But it is given that ab = ac => b = c, for all a, b, c in G.
    Therefore, (a_1)(a_j) = (a_1)(a_k) => a_j = ...

    Solution Summary

    This solution provides detailed steps explaining how to evaluate the given data as a group or not.