We shall define the right and
left indices of a subgroup G in a semigroup S with unit, and show that if G has
cancellation in S, and at least one of these indices is finite, then they are equal. If
only right cancellation holds, and the left index is finite, the right index will be either
less than the left index, or infinite. It will be shown by counterexamples that these
theorems are “best results.”
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