The index of a group in a semigroup
George M. Bergman · Pacific Journal of Mathematics · 1971
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."Let S be a semigroup with unit and G a subgroup of S with the same unit.Let [S: G]^ denote the cardinality of the set of right cosets xG of G in S, and [S: G\/ the cardinality of the set of left cosets.These are equal when S is a group, because the operation x h-> x~ι gives an anti-automorphism of S and of G.For S any finite semigroup, we see by a counting argument that these two indices will be equal if G has left and right cancellation in S: