Abstract

Among the twenty-three mathematical problems that Hilbert posed at the Second International Congress of Mathematicians in Paris in 1900 (see [10]), the fifth problem is concerned with Lie’s theory of transformation groups, and in its second part deals with what Hilbert calls “infinite groups,” but which are not groups in the modern use of the term. The questions in this second part of the fifth problem concern functional equations and difference equations and have, for example, connections with the work of N.H. Abel. These questions lie completely outside the theory of transformation groups, and we do not discuss them here any further. We refer the reader to [1] for the “state of the art” of 1989 in this area of research. We recall that a topological transformation group consists of a topological group G, a topological space X, and a continuous action of G on X, that is, a continuous map

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