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    A Set with an assoc. binary operation which is not a group

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    On the set Z of integer numbers we are given a binary operation o as follows:

    n o m = n + m + nm, for any n, m E Z.
    Prove that this operation is associative, however the pair (Z, o) is not a group.

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    Solution Preview

    Let us consider
    (n o m) o k = (n + m + nm) o k = n + m + nm + k + (n + m + nm)k = n + m + nm + nk + mk + nm +nmk

    for any n, m, k E Z.

    We have

    n o (m o k) = n o (m + k + mk) = n + m + k + mk + n(m + k + mk) = n + ...

    Solution Summary

    We first verify that the given operation is associative. Then we show that 0 is the unit with respect to this operation. Further, we show that 1 does not have an inverse element. We conclude that the given set with the given binary operation is not a group.