Finite Bruck loops
Authors:
Michael Aschbacher, Michael K. Kinyon and J. D. Phillips
Journal:
Trans. Amer. Math. Soc. 358 (2006), 3061-3075
MSC (2000):
Primary 20N05
DOI:
https://doi.org/10.1090/S0002-9947-05-03778-5
Published electronically:
September 22, 2005
MathSciNet review:
2216258
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Bruck loops are Bol loops satisfying the automorphic inverse property. We prove a structure theorem for finite Bruck loops 
, showing that 
is essentially the direct product of a Bruck loop of odd order with a 
-element Bruck loop. The former class of loops is well understood. We identify the minimal obstructions to the conjecture that all finite 
-element Bruck loops are 
-loops, leaving open the question of whether such obstructions actually exist.
- [A1] Michael Aschbacher, Near subgroups of finite groups, J. Group Theory 1 (1998), no. 2, 113–129. MR 1614316, https://doi.org/10.1515/jgth.1998.005
- [A2] M. Aschbacher, On Bol loops of of exponent 2 (to appear in J. Algebra).
- [Ba] Reinhold Baer, Nets and groups, Trans. Amer. Math. Soc. 46 (1939), 110–141. MR 0000035, https://doi.org/10.1090/S0002-9947-1939-0000035-5
- [Br] Richard Hubert Bruck, A survey of binary systems, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 20. Reihe: Gruppentheorie, Springer Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0093552
- [FT] Walter Feit and John G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029. MR 0166261
- [FGT] Michael Aschbacher, Finite group theory, Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 1986. MR 895134
- [G1] George Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403–420. MR 0202822, https://doi.org/10.1016/0021-8693(66)90030-5
- [G2] George Glauberman, On loops of odd order, J. Algebra 1 (1964), 374–396. MR 0175991, https://doi.org/10.1016/0021-8693(64)90017-1
- [G3] George Glauberman, On loops of odd order. II, J. Algebra 8 (1968), 393–414. MR 0222198, https://doi.org/10.1016/0021-8693(68)90050-1
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Additional Information
Michael Aschbacher
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Michael K. Kinyon
Affiliation:
Department of Mathematical Sciences, Indiana University South Bend, South Bend, Indiana 46634
J. D. Phillips
Affiliation:
Department of Mathematics and Computer Science, Wabash College, Crawfordsville, Indiana 47933
DOI:
https://doi.org/10.1090/S0002-9947-05-03778-5
Received by editor(s):
December 15, 2003
Received by editor(s) in revised form:
June 29, 2004
Published electronically:
September 22, 2005
Additional Notes:
The first author was partially supported by NSF-0203417
Article copyright:
© Copyright 2005
American Mathematical Society