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Finite Subgroups of F 4 (C) and E 6 (C)
Author(s) -
Cohen AM,
Wales DB
Publication year - 1997
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611597000051
Subject(s) - mathematics , combinatorics , isomorphism (crystallography) , simple group , lie group , classification of finite simple groups , simple lie group , simple (philosophy) , group of lie type , discrete mathematics , pure mathematics , group theory , crystallography , philosophy , chemistry , epistemology , crystal structure
The isomorphism types of finite Lie primitive subgroups of the complex Lie groups E 6 (C) and F 4 (C) are determined. Here, we call a finite subgroup of a complex Lie group G Lie primitive if it is not contained in a proper closed subgroup of G of positive dimension. Induction can be used to investigate subgroups which are not Lie primitive. Some additional information is provided, such as the characters of these finite subgroups on some small‐dimensional modules for the Lie groups. In studying these groups, we mainly use two rational linear representations of the universal covering groupE ~of E 6 (C), namely a 27‐dimensional module (there are two inequivalent ones), denoted by K, and the adjoint module. In particular, we make heavy use of the characters ofE ~on these modules. The group F 4 (C) occurs inE ~as the stabilizer subgroup of a vector in K. The finite simple groups of which a perfect central extension occurs in F 4 (C) or E 6 (C) are: via G 2 : Alt 5 , Alt 6 , L (2,7), L (2,8), L (2,13), U (3,3), via F 4 : Alt 7 , Alt 8 , Alt 9 , L (2,17), L (2,25), L (2,27), L (3,3), 3 D 4 ( 2 ) , U (4,2), O (7,2), O + (8,2), via E 6 : Alt 10 , Alt 11 , L (2, 11), L (2,19), L (3,4), U (4,3), 2 F 4( 2 ) ′ , M 11 , J 2 . This list has been found using the classification of the finite simple groups. On the basis of this list, the finite Lie primitive subgroups are found to be either the normalizers of one of these subgroups or of one of the two elementary abelian 3 ‐groups found by Alekseevskii. 1991 Mathematics Subject Classification : 20K47, 20G40, 17B45, 20C10.