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Composition Factors from the Group Ring and Artin's Theorem on Orders of Simple Groups
Author(s) -
Kimmerle Wolfgang,
Lyons Richard,
Sandling Robert,
Teague David N.
Publication year - 1990
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/plms/s3-60.1.89
Subject(s) - mathematics , simple group , simple (philosophy) , classification of finite simple groups , group (periodic table) , isomorphism (crystallography) , finite group , ring (chemistry) , mathematical proof , pure mathematics , order (exchange) , artin l function , simple module , group of lie type , algebra over a field , group theory , conductor , geometry , philosophy , chemistry , organic chemistry , epistemology , crystal structure , finance , economics , crystallography
The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a finite simple group is determined by its order with the same two types of exception which Artin found. Its proof, given here in detail, adapts and makes explicit certain functions of a natural number variable which Artin used implicitly. These functions contribute to the argument through a series of tables which supply their values for the orders of finite simple groups.