THE KNOWN SYSTEMS OF SIMPLE GROUPS

AND THEIR INTER-ISOMORPHISMS;

B y L.-E. DICKSON (Chicago).

The widespread interest laken ia the group-theory seems to be due to ils ready application lo many problems in geometry, function-theory and îiumber-theory, as well as in the theory of algebraic and differential équations. When a problem lias been exhibited in group phraseology, the possibility of a solation of a certain character or the exact nature of its inhérent diffîculties is determined by a study of the group of the problem. For example, the question of the solution of an algebraic équation b}r radicals or of the intégration of a differential équation by quadratures is answered by a knowledge of the group of the équation. Ultimately it is a question of the structure of the group, as determined by the chain of simple groups arising from its décomposition. As the chemist analyzes a compound to détermine the ultimate éléments composing it, so the group-theorist décomposés the'group of a given problem inlo a chain of simple groups. VYhile the chemist is concerned with about seventy éléments and their various compounds, the mathematician lias to study a universe formed from an infinité number of éléments, the simple groups.

Much labor has been expended in the détermination of simple groups. For continuons grou ps of a fini te number of parameters, the problem has been completely solved by Killing and Cartan, with the resuit that ail such simple groups, aside from five isolated ones, belong to the Systems investigated by Sophus Lie, viz., the general projective group, the projective group of a linear complex, and the projective group leaving invariant a non-degenerate surface of the second order. The corresponding problem for infinité continuons groups remains to be solved.

With regard to finite simple groups, the problem has been attacked in two directions. Hôlder, Gole, Burnside, Ling and Miller hâve shown that the only simple groups of composite orders less than 2000 are the

15

---

Le texte affiché peut comporter un certain nombre d'erreurs. En effet, le mode texte de ce document a été généré de façon automatique par un programme de reconnaissance optique de caractères (OCR). Le taux de reconnaissance estimé pour cette page est de 94,54 %.

La langue de reconnaissance de l'OCR est le Français.

---