Year: 1905
Type: article
Abstract: THEOREM VI. §-In the special cases in which n = 2, 3, 4, 5, 6, the abstract group (x 2 ' ' (2 2 »-i)> simply isomorphic with the group LF [2, 2 n ], * Relations (1), T-= / , [A, /x], \, ix any marks such that \n^fc I, constitute a set of generational relations for G.This is a special case of a more general theorem valid for any field due to Moore.See Proc.London Math.Soc, Vol.xxiv., p. 293.and Dickson'B Linear Groups, p. 300.Note that, when \ = 0 or 1, [A, M] reduces to (»S t T) 3 = I, and, when A = -1, [A, /u] reduces to ( 15).t For the special cases in which p n = 9, 25, 27, 125, 243, Prof. Dickson has proved that (1), (2), (3) constitute a set of generational relations for G. n .p 2»_ 1 \> i > > 2 , loc.cit.The proofs for the cases in which p" = 125, 243 have not been published.X This theorem is due to de Seguier, Journal de Mathematiques, Tome vm., p. 253.§ The set of generational relations (E) is due to Prof. Dickson.He has proved Theorem VI. for u = 2, 3, 4. See Proc.London Math.Soc, Vol.xxxv., p. 306 and p. 443; Bull.Amer.Math.Soc, Vol.IX., pp.194-204.For n = 2, the set (E) reduces to A» = I, B* = I, {AB)* = / ." Traite den Substitutions, p. 32.
Author: W. H. Bussey
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2021968911