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226 SECONDE PARTIE. — CONFÉRENCES ET COMMUNICATIONS. — SECTION I.

previously known simple groups of orders 60, 168, 36o, 5o4, 660, 1092. On tlie olher hand, various infinité Systems of finite simple groups hâve been determined. ïhe cyclic groups of prime orders and the allernating group on n letters (nf> 4) hâve long been recognized as simple groups. The other known Systems of finite simple groups bave been discovered in the study of linear groups. Four Systems were found by Jordan (*) in bis study of the general linear, the abelian, and the two hypoabelian groups, the field of reference being the set of residues of integers' with respect to a prime modulusp. Generalizations may be made by employing the Galois field (2) of order p'1 (designated GF[/?"]), composed of the p11 Galois complexes formed with a root of a congruence of degree n irredu-cible modulo p. Groups of linear substitutions in a Galois field were studied by Betti, Mathieu and Jordan; but the structure of such groups lias been determined only in the past decade. The simplicity of the group of unary linear fractional substitutions in a Galois field was first proved by Moore (3) and shorlly afterwards by Burnside (4). The complété generalization of Jordan’s four Systems of simple groups and the détermination of three new triply-infinite Systems hâve been made by the writer.

Aside frorn the cyclic and alternating groups, the known Systems of finite simple groups hâve been derived as quotient-groups in the sériés of composition of the following linear groups :

1. The group of ail linear homogeneous substitutions ( ;;)

m

f,: = ij\j G'=r>2. ....m),

7 = 1

on m indices with coefficients in the GF[/i"]. The quotient-group may be exhibited concretely as the group of linear fractional substitutions of déterminant unity on m — 1 indices.

2. The abelian linear group (°) of ail linear homogeneous substitutions

(>) Traité des substitutions.

(2) Galois, Œuvres mathématiques, Paris, 1897. — Moore, Congress Mathe-matical Papers, p. 208-242.

(3) Bulletin Amer. Math. Soc., Dec. 1893 ; in full in the Congress Math. Papers. (v) Proc. Lond. Math. Soc., vol. XXV, Feb. 18g4*

(5) Jordan, Traité-, n05 119, 169. — Dickson, Annals of Matliematics, p. 161 — 183, 1897; University of Chicago Record, Aug., 1896.

(®) Quarterly Journal of Math., p. 169-178, 1897; p. 383-384, 1899.

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