Embedding semigroups in groups: not as simple as it might seem

We consider the investigation of the embedding of semigroups in groups, a problem which spans the early-twentieth-century development of abstract algebra. Although this is a simple problem to state, it has proved rather harder to solve, and its apparent simplicity caused some of its would-be solvers to go awry. We begin with the analogous problem for rings, as dealt with by Ernst Steinitz, B. L. van der Waerden and Øystein Ore. After disposing of A. K. Sushkevich’s erroneous contribution in this area, we present A. I. Maltsev’s example of a cancellative semigroup which may not be embedded in a group, which showed for the first time that such an embedding is not possible in general. We then look at the various conditions that were derived for such an embedding to take place: the sufficient conditions of Paul Dubreil and others, and the necessary and sufficient conditions obtained by A. I. Maltsev, Vlastimil Pták and Joachim Lambek. We conclude with some comments on the place of this problem within the theory of semigroups, and also within abstract algebra more generally.