The finite Bruck loops

JOURNAL OF ALGEBRA

卷:330

Article

Baumeister, B.¹  Stein, A.¹ 

[1] Free Univ Berlin, Fachbereich Math & Informat, D-1000 Berlin, Germany

关键词: Finite Bruck loop;    Finite Bol loop;    Conjugacy classes of involutions;    Transversal;    Twisted subgroup;    Loop envelope;    Loop folder;    Fermat prime;    Structure of Bruck loop;    Sylow's Theorem;    Lagrange's Theorem;    Hall's Theorem;    A(r)-loop;   

DOI  :  10.1016/j.jalgebra.2010.11.017

来源: Elsevier

【 摘 要 】

We continue the work of Aschbacher. Kinyon and Phillips (2006) [AKP06] as well as of Glauberman (1964, 1968) [G64,G68] by describing the structure of the finite Bruck loops. We show that a finite Bruck loop X is the direct product of a Bruck loop of odd order with either a soluble Bruck loop of 2-power order or a product of loops related to the groups PSL(2)(q), q = 9 or q >= 5 a Fermat prime. The latter possibility does occur as is shown in Nagy (2008) [N08] and Baumeister and Stein (2010) [BS10]. As corollaries we obtain versions of Sylow's, Lagrange's and Hall's Theorems for loops. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free