Abstract
This chapter is devoted to Cartan's structural theory of infinite continuous groups. We thoroughly analyze its genesis and subsequent development. Cartan's approach to infinite continuous groups was characterized by a sharp break with the past tradition dating back to Lie, Engel, Medolaghi, and Vessiot. Such a discontinuity involved at least two aspects of the theory, the technical tools employed and the priorities of the theory itself. On the technical side, Cartan made a great profit of his theory of exterior differential systems, namely of his existence and uniqueness results for not integrable Pfaffian systems. As far as priorities were concerned, the Cartan theory was marked by a radical change of perspective: the emphasis was not put anymore on the problem of determining all infinite continuous groups of transformations in a given number of variables; rather, Cartan considered to be essential developing a structural theory of such groups in which the notion of isomorphism played a central role. Cartan's reinterpretation of Lie's equivalence theory is singled out as the main driving force in the genetical process leading to Cartan's structural theory of infinite groups.
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